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When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For
example, consider the product The answer is not in proper scientific notation
because 35 is greater than 10. Consider 35 as That adds a ten to the exponent of the answer.
EXAMPLE 12
Using Scientific Notation
Perform the operations and write the answer in scientific notation.
ⓐ ⓑ ⓒ
ⓓ ⓔSolution
ⓐ
ⓑ
ⓒ
ⓓ
ⓔ
TRY IT #12 Perform the operations and write the answer in scientific notation.
1.2 • Exponents and Scientific Notation 35
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ⓐ ⓑ
ⓒ ⓓ
ⓔ
EXAMPLE 13
Applying Scientific Notation to Solve Problems
In April 2014, the population of the United States was about 308,000,000 people. The national debt was about
$17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the
amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.
Solution
The population was
The national debt was
To find the amount of debt per citizen, divide the national debt by the number of citizens.
The debt per citizen at the time was about or $57,000.
TRY IT #13 An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures
approximately 0.000008 m long. Write each number in scientific notation and find the total length
if the cells were laid end-to-end. Write the answer in both scientific and standard notations.
MEDIA
Access these online resources for additional instruction and practice with exponents and scientific notation.
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Algebra-and-Trigonometry-2e-WEB.pdf#page=45&chunk=101
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MEDIA
Access these online resources for additional instruction and practice with exponents and scientific notation.
Exponential Notation (http://openstax.org/l/exponnot)
Properties of Exponents (http://openstax.org/l/exponprops)
Zero Exponent (http://openstax.org/l/zeroexponent)
Simplify Exponent Expressions (http://openstax.org/l/exponexpres)
Quotient Rule for Exponents (http://openstax.org/l/quotofexpon)
Scientific Notation (http://openstax.org/l/scientificnota)
Converting to Decimal Notation (http://openstax.org/l/decimalnota)
1.2 SECTION EXERCISES
Verbal
1. Is the same as
Explain.
2. When can you add two
exponents?
3. What is the purpose of
scientific notation?
4. Explain what a negative
exponent does.
36 1 • Prerequisites
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Numeric
For the following exercises, simplify the given expression. Write answers with positive exponents.
5. 6. 7.
8. 9. 10.
11. 12. 13.
14.
For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive
exponents.
15. 16. 17.
18. 19. 20.
For the following exercises, express the decimal in scientific notation.
21. 0.0000314 22. 148,000,000
For the following exercises, convert each number in scientific notation to standard notation.
23. 24.
Algebraic
For the following exercises, simplify the given expression. Write answers with positive exponents.
25. 26. 27.
28. 29. 30.
31. 32. 33.
34. 35. 36.
37. 38. 39.
40. 41. 42.
1.2 • Exponents and Scientific Notation 37
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43.
Real-World Applications
44. To reach escape velocity, a
rocket must travel at the
rate of ft/min.
Rewrite the rate in
standard notation.
45. A dime is the thinnest coin
in U.S. currency. A dime’s
thickness measures
m. Rewrite the
number in standard
notation.
46. The average distance
between Earth and the Sun
is 92,960,000 mi. Rewrite
the distance using scientific
notation.
47. A terabyte is made of
approximately
1,099,500,000,000 bytes.
Rewrite in scientific
notation.
48. The Gross Domestic
Product (GDP) for the
United States in the first
quarter of 2014 was
Rewrite
the GDP in standard
notation.
49. One picometer is
approximately
in. Rewrite
this length using standard
notation.
50. The value of the services
sector of the U.S. economy
in the first quarter of 2012
was $10,633.6 billion.
Rewrite this amount in
scientific notation.
Technology
For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.
51. 52.
Extensions
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Rewrite this amount in
scientific notation.
Technology
For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.
51. 52.
Extensions
For the following exercises, simplify the given expression. Write answers with positive exponents.
53. 54. 55.
56. 57. 58. Avogadro’s constant is
used to calculate the
number of particles in a
mole. A mole is a basic unit
in chemistry to measure
the amount of a substance.
The constant is
Write
Avogadro’s constant in
standard notation.
38 1 • Prerequisites
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59. Planck’s constant is an
important unit of measure
in quantum physics. It
describes the relationship
between energy and
frequency. The constant is
written as
Write
Planck’s constant in
standard notation.
1.3 Radicals and Rational Exponents
Learning Objectives
In this section, you will:
Evaluate square roots.
Use the product rule to simplify square roots.
Use the quotient rule to simplify square roots.
Add and subtract square roots.
Rationalize denominators.
Use rational roots.
A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to
be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of
ladder needed, we can draw a right triangle as shown in Figure 1, and use the Pythagorean Theorem.
Figure 1
Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we
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Figure 1
Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we
need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.
Evaluating Square Roots
When the square root of a number is squared, the result is the original number. Since the square root of is
The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo
squaring, we take the square root.
In general terms, if is a positive real number, then the square root of is a number that, when multiplied by itself,
gives The square root could be positive or negative because multiplying two negative numbers gives a positive
number. The principal square root is the nonnegative number that when multiplied by itself equals The square root
obtained using a calculator is the principal square root.
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number. The principal square root is the nonnegative number that when multiplied by itself equals The square root
obtained using a calculator is the principal square root.
The principal square root of is written as The symbol is called a radical, the term under the symbol is called the
radicand, and the entire expression is called a radical expression.
1.3 • Radicals and Rational Exponents 39
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...
Principal Square Root
The principal square root of is the nonnegative number that, when multiplied by itself, equals It is written as a
radical expression, with a symbol called a radical over the term called the radicand:
Q&A Does
No. Although both and are the radical symbol implies only a nonnegative root, the principal
square root. The principal square root of 25 is
EXAMPLE 1
Evaluating Square Roots
Evaluate each expression.
ⓐ ⓑ ⓒ ⓓ
Solution
ⓐ because ⓑ because and
ⓒ because ⓓ because and
Q&A For can we find the square roots before adding?
No. This is not equivalent to The order of operations
requires us to add the terms in the radicand before finding the square root.
TRY IT #1 Evaluate each expression.
ⓐ ⓑ ⓒ ⓓ
Using the Product Rule to Simplify Square Roots
To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several
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ⓐ ⓑ ⓒ ⓓ
Using the Product Rule to Simplify Square Roots
To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several
properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the
product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into
the product of two separate rational expressions. For instance, we can rewrite as We can also use the
product rule to express the product of multiple radical expressions as a single radical expression.
The Product Rule for Simplifying Square Roots
If and are nonnegative, the square root of the product is equal to the product of the square roots of and
HOW TO
Given a square root radical expression, use the product rule to simplify it.
40 1 • Prerequisites
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...
1. Factor any perfect squares from the radicand.
2. Write the radical expression as a product of radical expressions.
3. Simplify.
EXAMPLE 2
Using the Product Rule to Simplify Square Roots
Simplify the radical expression.
ⓐ ⓑSolution
ⓐ
ⓑ
TRY IT #2 Simplify
HOW TO
Given the product of multiple radical expressions, use the product rule to combine them into one radical
expression.
1. Express the product of multiple radical expressions as a single radical expression.
2. Simplify.
EXAMPLE 3
Using the Product Rule to Simplify the Product of Multiple Square Roots
Simplify the radical expression.
Solution
TRY IT #3 Simplify assuming
Using the Quotient Rule to Simplify Square Roots
Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of
a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to
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a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to
separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately.
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We can rewrite as
The Quotient Rule for Simplifying Square Roots
The square root of the quotient is equal to the quotient of the square roots of and where
HOW TO
Given a radical expression, use the quotient rule to simplify it.
1. Write the radical expression as the quotient of two radical expressions.
2. Simplify the numerator and denominator.
EXAMPLE 4
Using the Quotient Rule to Simplify Square Roots
Simplify the radical expression.
Solution
TRY IT #4 Simplify
EXAMPLE 5
Using the Quotient Rule to Simplify an Expression with Two Square Roots
Simplify the radical expression.
Solution
TRY IT #5 Simplify
42 1 • Prerequisites
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...
Adding and Subtracting Square Roots
We can add or subtract radical expressions only when they have the same radicand and when they have the same radical
type such as square roots. For example, the sum of and is However, it is often possible to simplify radical
expressions, and that may change the radicand. The radical expression can be written with a in the radicand, as
so
HOW TO
Given a radical expression requiring addition or subtraction of square roots, simplify.
1. Simplify each radical expression.
2. Add or subtract expressions with equal radicands.
EXAMPLE 6
Adding Square Roots
Add
Solution
We can rewrite as According the product rule, this becomes The square root of is 2, so the
expression becomes which is Now the terms have the same radicand so we can add.
TRY IT #6 Add
EXAMPLE 7
Subtracting Square Roots
Subtract
Solution
Rewrite each term so they have equal radicands.
Now the terms have the same radicand so we can subtract.
TRY IT #7 Subtract
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TRY IT #6 Add
EXAMPLE 7
Subtracting Square Roots
Subtract
Solution
Rewrite each term so they have equal radicands.
Now the terms have the same radicand so we can subtract.
TRY IT #7 Subtract
1.3 • Radicals and Rational Exponents 43
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...
...
Rationalizing Denominators
When an expression involving square root radicals is written in simplest form, it will not contain a radical in the
denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the
denominator.
We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to
change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions,
multiply by the form of 1 that will eliminate the radical.
For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the
denominator is multiply by
For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and
denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the
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denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the
denominator. If the denominator is then the conjugate is
HOW TO
Given an expression with a single square root radical term in the denominator, rationalize the denominator.
a. Multiply the numerator and denominator by the radical in the denominator.
b. Simplify.
EXAMPLE 8
Rationalizing a Denominator Containing a Single Term
Write in simplest form.
Solution
The radical in the denominator is So multiply the fraction by Then simplify.
TRY IT #8 Write in simplest form.
HOW TO
Given an expression with a radical term and a constant in the denominator, rationalize the denominator.
1. Find the conjugate of the denominator.
2. Multiply the numerator and denominator by the conjugate.
3. Use the distributive property.
4. Simplify.
44 1 • Prerequisites
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EXAMPLE 9
Rationalizing a Denominator Containing Two Terms
Write in simplest form.
Solution
Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate
of is Then multiply the fraction by
TRY IT #9 Write in simplest form.
Using Rational Roots
Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more.
Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective
power functions. These functions can be useful when we need to determine the number that, when raised to a certain
power, gives a certain number.
Understanding nth Roots
Suppose we know that We want to find what number raised to the 3rd power is equal to 8. Since we say
that 2 is the cube root of 8.
The nth root of is a number that, when raised to the nth power, gives For example, is the 5th root of
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that 2 is the cube root of 8.
The nth root of is a number that, when raised to the nth power, gives For example, is the 5th root of
because If is a real number with at least one nth root, then the principal nth root of is the number
with the same sign as that, when raised to the nth power, equals
The principal nth root of is written as where is a positive integer greater than or equal to 2. In the radical
expression, is called the index of the radical.
Principal th Root
If is a real number with at least one nth root, then the principal nth root of written as is the number with
the same sign as that, when raised to the nth power, equals The index of the radical is
EXAMPLE 10
Simplifying nth Roots
Simplify each of the following:
ⓐ ⓑ ⓒ ⓓ
Solution
ⓐ because
ⓑ First, express the product as a single radical expression. because
ⓒ
1.3 • Radicals and Rational Exponents 45
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ⓓ
TRY IT #10 Simplify.
ⓐ ⓑ ⓒ
Using Rational Exponents
Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents.
The index must be a positive integer. If the index is even, then cannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in
lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator
tells us the root.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.
Rational Exponents
Rational exponents are another way to express principal nth roots. The general form for converting between a radical
expression with a radical symbol and one with a rational exponent is
HOW TO
Given an expression with a rational exponent, write the expression as a radical.
1. Determine the power by looking at the numerator of the exponent.
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HOW TO
Given an expression with a rational exponent, write the expression as a radical.
1. Determine the power by looking at the numerator of the exponent.
2. Determine the root by looking at the denominator of the exponent.
3. Using the base as the radicand, raise the radicand to the power and use the root as the index.
EXAMPLE 11
Writing Rational Exponents as Radicals
Write as a radical. Simplify.
Solution
The 2 tells us the power and the 3 tells us the root.
We know that because Because the cube root is easy to find, it is easiest to find the cube root
before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.
46 1 • Prerequisites
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TRY IT #11 Write as a radical. Simplify.
EXAMPLE 12
Writing Radicals as Rational Exponents
Write using a rational exponent.
Solution
The power is 2 and the root is 7, so the rational exponent will be We get Using properties of exponents, we get
TRY IT #12 Write using a rational exponent.
EXAMPLE 13
Simplifying Rational Exponents
Simplify:
ⓐ ⓑ
Solution
ⓐ
ⓑ
TRY IT #13 Simplify
MEDIA
Access these online resources for additional instruction and practice with radicals and rational exponents.
Radicals (http://openstax.org/l/introradical)
Rational Exponents (http://openstax.org/l/rationexpon)
Simplify Radicals (http://openstax.org/l/simpradical)
Rationalize Denominator (http://openstax.org/l/rationdenom)
1.3 • Radicals and Rational Exponents 47
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1.3 SECTION EXERCISES
Verbal
1. What does it mean when a
radical does not have an
index? Is the expression
equal to the radicand?
Explain.
2. Where would radicals come
in the order of operations?
Explain why.
3. Every number will have two
square roots. What is the
principal square root?
4. Can a radical with a negative
radicand have a real square
root? Why or why not?
Numeric
For the following exercises, simplify each expression.
5. 6. 7.
8. 9. 10.
11. 12. 13.
14. 15. 16.
17. 18. 19.
20. 21. 22.
23. 24. 25.
26. 27. 28.
29. 30. 31.
32. 33. 34.
48 1 • Prerequisites
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Algebraic
For the following exercises, simplify each expression.
35. 36. 37.
38. 39. 40.
41. 42. 43.
44. 45. 46.
47. 48. 49.
50. 51. 52.
53. 54. 55.
56. 57. 58.
59. 60. 61.
62. 63. 64.
Real-World Applications
65. A guy wire for a suspension bridge runs from the
ground diagonally to the top of the closest pylon
to make a triangle. We can use the Pythagorean
Theorem to find the length of guy wire needed.
The square of the distance between the wire on
the ground and the pylon on the ground is 90,000
feet. The square of the height of the pylon is
160,000 feet. So the length of the guy wire can be
found by evaluating What is
the length of the guy wire?
66. A car accelerates at a rate of where
t is the time in seconds after the car moves from
rest. Simplify the expression.
1.3 • Radicals and Rational Exponents 49
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Extensions
For the following exercises, simplify each expression.
67. 68. 69.
70. 71. 72.
73.
1.4 Polynomials
Learning Objectives
In this section, you will:
Identify the degree and leading coefficient of polynomials.
Add and subtract polynomials.
Multiply polynomials.
Use FOIL to multiply binomials.
Perform operations with polynomials of several variables.
Maahi is building a little free library (a small house-shaped book repository), whose front is in the shape of a square
topped with a triangle. There will be a rectangular door through which people can take and donate books. Maahi wants
to find the area of the front of the library so that they can purchase the correct amount of paint. Using the
measurements of the front of the house, shown in Figure 1, we can create an expression that combines several variable
terms, allowing us to solve this problem and others like it.
Figure 1
First find the area of the square in square feet.
Then find the area of the triangle in square feet.
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terms, allowing us to solve this problem and others like it.
Figure 1
First find the area of the square in square feet.
Then find the area of the triangle in square feet.
Next find the area of the rectangular door in square feet.
50 1 • Prerequisites
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...
The area of the front of the library can be found by adding the areas of the square and the triangle, and then subtracting
the area of the rectangle. When we do this, we get or ft2.
In this section, we will examine expressions such as this one, which combine several variable terms.
Identifying the Degree and Leading Coefficient of Polynomials
The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a
variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as
is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or
fractions. Each product such as is a term of a polynomial. If a term does not contain a variable, it is called
a constant.
A polynomial containing only one term, such as is called a monomial. A polynomial containing two terms, such as
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a constant.
A polynomial containing only one term, such as is called a monomial. A polynomial containing two terms, such as
is called a binomial. A polynomial containing three terms, such as is called a trinomial.
We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial.
The term with the highest degree is called the leading term because it is usually written first. The coefficient of the
leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say
that it is in standard form.
Polynomials
A polynomial is an expression that can be written in the form
Each real number ai is called a coefficient. The number that is not multiplied by a variable is called a constant.
Each product is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called
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Each product is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called
the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the
leading coefficient.
HOW TO
Given a polynomial expression, identify the degree and leading coefficient.
1. Find the highest power of x to determine the degree.
2. Identify the term containing the highest power of x to find the leading term.
3. Identify the coefficient of the leading term.
EXAMPLE 1
Identifying the Degree and Leading Coefficient of a Polynomial
For the following polynomials, identify the degree, the leading term, and the leading coefficient.
ⓐ ⓑ ⓒSolution
ⓐ The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, The
leading coefficient is the coefficient of that term,
ⓑ The highest power of t is so the degree is The leading term is the term containing that degree, The
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leading coefficient is the coefficient of that term,
ⓑ The highest power of t is so the degree is The leading term is the term containing that degree, The
leading coefficient is the coefficient of that term,
1.4 • Polynomials 51
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...
ⓒ The highest power of p is so the degree is The leading term is the term containing that degree, The
leading coefficient is the coefficient of that term,
TRY IT #1 Identify the degree, leading term, and leading coefficient of the polynomial
Adding and Subtracting Polynomials
We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to
the same exponents. For example, and are like terms, and can be added to get but and are not
like terms, and therefore cannot be added.
HOW TO
Given multiple polynomials, add or subtract them to simplify the expressions.
1. Combine like terms.
2. Simplify and write in standard form.
EXAMPLE 2
Adding Polynomials
Find the sum.
Solution
Analysis
We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given
along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the
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along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the
graphs. Using different windows can make the expressions seem equivalent when they are not.
TRY IT #2 Find the sum.
EXAMPLE 3
Subtracting Polynomials
Find the difference.
Solution
Analysis
Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial
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...
to the first.
TRY IT #3 Find the difference.
Multiplying Polynomials
Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive
property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like
terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow
patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a
variety of ways to multiply polynomials.
Multiplying Polynomials Using the Distributive Property
To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of
the polynomial. We can distribute the in to obtain the equivalent expression When multiplying
polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second.
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polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second.
We then add the products together and combine like terms to simplify.
HOW TO
Given the multiplication of two polynomials, use the distributive property to simplify the expression.
1. Multiply each term of the first polynomial by each term of the second.
2. Combine like terms.
3. Simplify.
EXAMPLE 4
Multiplying Polynomials Using the Distributive Property
Find the product.
Solution
Analysis
We can use a table to keep track of our work, as shown in Table 1. Write one polynomial across the top and the other
down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the
terms together, combine like terms, and simplify.
Table 1
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Table 1
TRY IT #4 Find the product.
Using FOIL to Multiply Binomials
A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the
first terms, the outer terms, the inner terms, and then the last terms of each binomial.
The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by
each term of the second binomial, and then combining like terms.
HOW TO
Given two binomials, use FOIL to simplify the expression.
1. Multiply the first terms of each binomial.
2. Multiply the outer terms of the binomials.
3. Multiply the inner terms of the binomials.
4. Multiply the last terms of each binomial.
5. Add the products.
6. Combine like terms and simplify.
EXAMPLE 5
Using FOIL to Multiply Binomials
Use FOIL to find the product.
Solution
Find the product of the first terms.
Find the product of the outer terms.
Find the product of the inner terms.
Find the product of the last terms.
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Use FOIL to find the product.
Solution
Find the product of the first terms.
Find the product of the outer terms.
Find the product of the inner terms.
Find the product of the last terms.
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TRY IT #5 Use FOIL to find the product.
Perfect Square Trinomials
Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square
trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of
these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s
look at a few perfect square trinomials to familiarize ourselves with the form.
Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of
each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms.
Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.
Perfect Square Trinomials
When a binomial is squared, the result is the first term squared added to double the product of both terms and the
last term squared.
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Perfect Square Trinomials
When a binomial is squared, the result is the first term squared added to double the product of both terms and the
last term squared.
HOW TO
Given a binomial, square it using the formula for perfect square trinomials.
1. Square the first term of the binomial.
2. Square the last term of the binomial.
3. For the middle term of the trinomial, double the product of the two terms.
4. Add and simplify.
EXAMPLE 6
Expanding Perfect Squares
Expand
Solution
Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two
terms.
Simplify.
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TRY IT #6 Expand
Difference of Squares
Another special product is called the difference of squares, which occurs when we multiply a binomial by another
binomial with the same terms but the opposite sign. Let’s see what happens when we multiply using the
FOIL method.
The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let’s look at a few
examples.
Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only
with the square of the first term minus the square of the last term.
Q&A Is there a special form for the sum of squares?
No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms
to disappear. There are no two binomials that multiply to equal a sum of squares.
Difference of Squares
When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the
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Difference of Squares
When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the
square of the first term minus the square of the last term.
HOW TO
Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of
squares.
1. Square the first term of the binomials.
2. Square the last term of the binomials.
3. Subtract the square of the last term from the square of the first term.
EXAMPLE 7
Multiplying Binomials Resulting in a Difference of Squares
Multiply
Solution
Square the first term to get Square the last term to get Subtract the square of the last term from
the square of the first term to find the product of
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TRY IT #7 Multiply
Performing Operations with Polynomials of Several Variables
We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of
the same rules apply when working with polynomials containing several variables. Consider an example:
EXAMPLE 8
Multiplying Polynomials Containing Several Variables
Multiply
Solution
Follow the same steps that we used to multiply polynomials containing only one variable.
TRY IT #8 Multiply
MEDIA
Access these online resources for additional instruction and practice with polynomials.
Adding and Subtracting Polynomials (http://openstax.org/l/addsubpoly)
Multiplying Polynomials (http://openstax.org/l/multiplpoly)
Special Products of Polynomials (http://openstax.org/l/specialpolyprod)
1.4 SECTION EXERCISES
Verbal
1. Evaluate the following
statement: The degree of a
polynomial in standard form
is the exponent of the
leading term. Explain why
the statement is true or
false.
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1.4 SECTION EXERCISES
Verbal
1. Evaluate the following
statement: The degree of a
polynomial in standard form
is the exponent of the
leading term. Explain why
the statement is true or
false.
2. Many times, multiplying two
binomials with two variables
results in a trinomial. This is
not the case when there is a
difference of two squares.
Explain why the product in
this case is also a binomial.
3. You can multiply
polynomials with any
number of terms and any
number of variables using
four basic steps over and
over until you reach the
expanded polynomial. What
are the four steps?
4. State whether the following
statement is true and
explain why or why not: A
trinomial is always a higher
degree than a monomial.
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Algebraic
For the following exercises, identify the degree of the polynomial.
5. 6. 7.
8. 9. 10.
For the following exercises, find the sum or difference.
11. 12.
13. 14.
15. 16.
For the following exercises, find the product.
17. 18. 19.
20. 21. 22.
23.
For the following exercises, expand the binomial.
24. 25. 26.
27. 28. 29.
30.
For the following exercises, multiply the binomials.
31. 32. 33.
34. 35. 36.
37.
For the following exercises, multiply the polynomials.
38. 39. 40.
41. 42. 43.
44. 45. 46.
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47. 48. 49.
50. 51. 52.
Real-World Applications
53. A developer wants to purchase a plot of land to
build a house. The area of the plot can be
described by the following expression:
where x is measured in meters.
Multiply the binomials to find the area of the plot
in standard form.
54. A prospective buyer wants to know how much
grain a specific silo can hold. The area of the floor
of the silo is The height of the silo is
where x is measured in feet. Expand the
square and multiply by the height to find the
expression that shows how much grain the silo
can hold.
Extensions
For the following exercises, perform the given operations.
55. 56. 57.
1.5 Factoring Polynomials
Learning Objectives
In this section, you will:
Factor the greatest common factor of a polynomial.
Factor a trinomial.
Factor by grouping.
Factor a perfect square trinomial.
Factor a difference of squares.
Factor the sum and difference of cubes.
Factor expressions using fractional or negative exponents.
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Factor by grouping.
Factor a perfect square trinomial.
Factor a difference of squares.
Factor the sum and difference of cubes.
Factor expressions using fractional or negative exponents.
Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The
lawn is the green portion in Figure 1.
Figure 1
The area of the entire region can be found using the formula for the area of a rectangle.
The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The
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two square regions each have an area of units2. The other rectangular region has one side of length
and one side of length giving an area of units2. So the region that must be
subtracted has an area of units2.
The area of the region that requires grass seed is found by subtracting units2. This area can also be
expressed in factored form as units2. We can confirm that this is an equivalent expression by multiplying.
Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of
methods that can be used to factor polynomial expressions.
Factoring the Greatest Common Factor of a Polynomial
When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that
divides evenly into both numbers. For instance, is the GCF of and because it is the largest number that divides
evenly into both and The GCF of polynomials works the same way: is the GCF of and because it is the
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evenly into both and The GCF of polynomials works the same way: is the GCF of and because it is the
largest polynomial that divides evenly into both and
When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients,
and then look for the GCF of the variables.
Greatest Common Factor
The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.
HOW TO
Given a polynomial expression, factor out the greatest common factor.
1. Identify the GCF of the coefficients.
2. Identify the GCF of the variables.
3. Combine to find the GCF of the expression.
4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.
EXAMPLE 1
Factoring the Greatest Common Factor
Factor
Solution
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5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.
EXAMPLE 1
Factoring the Greatest Common Factor
Factor
Solution
First, find the GCF of the expression. The GCF of 6, 45, and 21 is 3. The GCF of , and is . (Note that the GCF of a
set of expressions in the form will always be the exponent of lowest degree.) And the GCF of , and is .
Combine these to find the GCF of the polynomial, .
Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that
, and
Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.
Analysis
After factoring, we can check our work by multiplying. Use the distributive property to confirm that
TRY IT #1 Factor by pulling out the GCF.
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Factoring a Trinomial with Leading Coefficient 1
Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial
expressions can be factored. The polynomial has a GCF of 1, but it can be written as the product of the
factors and
Trinomials of the form can be factored by finding two numbers with a product of and a sum of The
trinomial for example, can be factored using the numbers and because the product of those numbers
is and their sum is The trinomial can be rewritten as the product of and
Factoring a Trinomial with Leading Coefficient 1
A trinomial of the form can be written in factored form as where and
Q&A Can every trinomial be factored as a product of binomials?
No. Some polynomials cannot be factored. These polynomials are said to be prime.
HOW TO
Given a trinomial in the form factor it.
1. List factors of
2. Find and a pair of factors of with a sum of
3. Write the factored expression
EXAMPLE 2
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HOW TO
Given a trinomial in the form factor it.
1. List factors of
2. Find and a pair of factors of with a sum of
3. Write the factored expression
EXAMPLE 2
Factoring a Trinomial with Leading Coefficient 1
Factor
Solution
We have a trinomial with leading coefficient and We need to find two numbers with a product of
and a sum of In the table below, we list factors until we find a pair with the desired sum.
Factors of Sum of Factors
14
2
Now that we have identified and as and write the factored form as
Analysis
We can check our work by multiplying. Use FOIL to confirm that
Q&A Does the order of the factors matter?
No. Multiplication is commutative, so the order of the factors does not matter.
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TRY IT #2 Factor
Factoring by Grouping
Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can
factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression
separately, and then factoring out the GCF of the entire expression. The trinomial can be rewritten as
using this process. We begin by rewriting the original expression as and then factor
each portion of the expression to obtain We then pull out the GCF of to find the factored
expression.
Factor by Grouping
To factor a trinomial in the form by grouping, we find two numbers with a product of and a sum of
We use these numbers to divide the term into the sum of two terms and factor each portion of the expression
separately, then factor out the GCF of the entire expression.
HOW TO
Given a trinomial in the form factor by grouping.
1. List factors of
2. Find and a pair of factors of with a sum of
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separately, then factor out the GCF of the entire expression.
HOW TO
Given a trinomial in the form factor by grouping.
1. List factors of
2. Find and a pair of factors of with a sum of
3. Rewrite the original expression as
4. Pull out the GCF of
5. Pull out the GCF of
6. Factor out the GCF of the expression.
EXAMPLE 3
Factoring a Trinomial by Grouping
Factor by grouping.
Solution
We have a trinomial with and First, determine We need to find two numbers with a
product of and a sum of In the table below, we list factors until we find a pair with the desired sum.
Factors of Sum of Factors
29
13
7
So and
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Analysis
We can check our work by multiplying. Use FOIL to confirm that
TRY IT #3 Factor
ⓐ ⓑ
Factoring a Perfect Square Trinomial
A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is
squared, the result is the square of the first term added to twice the product of the two terms and the square of the last
term.
We can use this equation to factor any perfect square trinomial.
Perfect Square Trinomials
A perfect square trinomial can be written as the square of a binomial:
HOW TO
Given a perfect square trinomial, factor it into the square of a binomial.
1. Confirm that the first and last term are perfect squares.
2. Confirm that the middle term is twice the product of
3. Write the factored form as
EXAMPLE 4
Factoring a Perfect Square Trinomial
Factor
Solution
Notice that and are perfect squares because and Then check to see if the middle term is
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3. Write the factored form as
EXAMPLE 4
Factoring a Perfect Square Trinomial
Factor
Solution
Notice that and are perfect squares because and Then check to see if the middle term is
twice the product of and The middle term is, indeed, twice the product: Therefore, the trinomial is
a perfect square trinomial and can be written as
TRY IT #4 Factor
Factoring a Difference of Squares
A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be
rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when
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the two factors are multiplied.
We can use this equation to factor any differences of squares.
Differences of Squares
A difference of squares can be rewritten as two factors containing the same terms but opposite signs.
HOW TO
Given a difference of squares, factor it into binomials.
1. Confirm that the first and last term are perfect squares.
2. Write the factored form as
EXAMPLE 5
Factoring a Difference of Squares
Factor
Solution
Notice that and are perfect squares because and The polynomial represents a difference of
squares and can be rewritten as
TRY IT #5 Factor
Q&A Is there a formula to factor the sum of squares?
No. A sum of squares cannot be factored.
Factoring the Sum and Difference of Cubes
Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be
factored, the sum of cubes can be factored into a binomial and a trinomial.
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Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be
factored, the sum of cubes can be factored into a binomial and a trinomial.
Similarly, the difference of cubes can be factored into a binomial and a trinomial, but with different signs.
We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of
each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.
The sign of the first 2 is the same as the sign between The sign of the term is opposite the sign between
And the sign of the last term, 4, is always positive.
Sum and Difference of Cubes
We can factor the sum of two cubes as
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We can factor the difference of two cubes as
HOW TO
Given a sum of cubes or difference of cubes, factor it.
1. Confirm that the first and last term are cubes, or
2. For a sum of cubes, write the factored form as For a difference of cubes, write the
factored form as
EXAMPLE 6
Factoring a Sum of Cubes
Factor
Solution
Notice that and are cubes because Rewrite the sum of cubes as
Analysis
After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be factored
further. However, the trinomial portion cannot be factored, so we do not need to check.
TRY IT #6 Factor the sum of cubes:
EXAMPLE 7
Factoring a Difference of Cubes
Factor
Solution
Notice that and are cubes because and Write the difference of cubes as
Analysis
Just as with the sum of cubes, we will not be able to further factor the trinomial portion.
TRY IT #7 Factor the difference of cubes:
Factoring Expressions with Fractional or Negative Exponents
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Just as with the sum of cubes, we will not be able to further factor the trinomial portion.
TRY IT #7 Factor the difference of cubes:
Factoring Expressions with Fractional or Negative Exponents
Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or
exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest
power. These expressions follow the same factoring rules as those with integer exponents. For instance, can
be factored by pulling out and being rewritten as
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EXAMPLE 8
Factoring an Expression with Fractional or Negative Exponents
Factor
Solution
Factor out the term with the lowest value of the exponent. In this case, that would be
TRY IT #8 Factor
MEDIA
Access these online resources for additional instruction and practice with factoring polynomials.
Identify GCF (http://openstax.org/l/findgcftofact)
Factor Trinomials when a Equals 1 (http://openstax.org/l/facttrinom1)
Factor Trinomials when a is not equal to 1 (http://openstax.org/l/facttrinom2)
Factor Sum or Difference of Cubes (http://openstax.org/l/sumdifcube)
1.5 SECTION EXERCISES
Verbal
1. If the terms of a polynomial
do not have a GCF, does that
mean it is not factorable?
Explain.
2. A polynomial is factorable,
but it is not a perfect square
trinomial or a difference of
two squares. Can you factor
the polynomial without
finding the GCF?
3. How do you factor by
grouping?
Algebraic
For the following exercises, find the greatest common factor.
4. 5. 6.
7. 8. 9.
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two squares. Can you factor
the polynomial without
finding the GCF?
3. How do you factor by
grouping?
Algebraic
For the following exercises, find the greatest common factor.
4. 5. 6.
7. 8. 9.
For the following exercises, factor by grouping.
10. 11. 12.
13. 14. 15.
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For the following exercises, factor the polynomial.
16. 17. 18.
19. 20. 21.
22. 23. 24.
25. 26. 27.
28. 29. 30.
31. 32. 33.
34. 35. 36.
For the following exercises, factor the polynomials.
37. 38. 39.
40. 41. 42.
43. 44. 45.
46. 47. 48.
49. 50.
Real-World Applications
For the following exercises, consider this scenario:
Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains
in one of the city’s parks. The park is a rectangle with an area of m2, as shown in the figure below. The
length and width of the park are perfect factors of the area.
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51. Factor by grouping to find
the length and width of the
park.
52. A statue is to be placed in
the center of the park. The
area of the base of the
statue is
Factor the area to find the
lengths of the sides of the
statue.
53. At the northwest corner of
the park, the city is going
to install a fountain. The
area of the base of the
fountain is
Factor the area to find the
lengths of the sides of the
fountain.
For the following exercise, consider the following scenario:
A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure
below. The flagpole will take up a square plot with area yd2.
54. Find the length of the base of the flagpole by
factoring.
Extensions
For the following exercises, factor the polynomials completely.
55. 56. 57.
58. 59.
1.6 Rational Expressions
Learning Objectives
In this section, you will:
Simplify rational expressions.
Multiply rational expressions.
Divide rational expressions.
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55. 56. 57.
58. 59.
1.6 Rational Expressions
Learning Objectives
In this section, you will:
Simplify rational expressions.
Multiply rational expressions.
Divide rational expressions.
Add and subtract rational expressions.
Simplify complex rational expressions.
A pastry shop has fixed costs of per week and variable costs of per box of pastries. The shop’s costs per week in
terms of the number of boxes made, is We can divide the costs per week by the number of boxes made to
determine the cost per box of pastries.
Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will
explore quotients of polynomial expressions.
Simplifying Rational Expressions
The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to
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rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the
denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational
expression shown.
We can factor the numerator and denominator to rewrite the expression.
Then we can simplify that expression by canceling the common factor
HOW TO
Given a rational expression, simplify it.
1. Factor the numerator and denominator.
2. Cancel any common factors.
EXAMPLE 1
Simplifying Rational Expressions
Simplify
Solution
Analysis
We can cancel the common factor because any expression divided by itself is equal to 1.
Q&A Can the term be cancelled in Example 1?
No. A factor is an expression that is multiplied by another expression. The term is not a factor of the
numerator or the denominator.
TRY IT #1 Simplify
Multiplying Rational Expressions
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No. A factor is an expression that is multiplied by another expression. The term is not a factor of the
numerator or the denominator.
TRY IT #1 Simplify
Multiplying Rational Expressions
Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the
numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the
product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying
rational expressions. We are often able to simplify the product of rational expressions.
HOW TO
Given two rational expressions, multiply them.
1. Factor the numerator and denominator.
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2. Multiply the numerators.
3. Multiply the denominators.
4. Simplify.
EXAMPLE 2
Multiplying Rational Expressions
Multiply the rational expressions and show the product in simplest form:
Solution
TRY IT #2 Multiply the rational expressions and show the product in simplest form:
Dividing Rational Expressions
Division of rational expressions works the same way as division of other fractions. To divide a rational expression by
another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would
rewrite as the product Once the division expression has been rewritten as a multiplication expression,
we can multiply as we did before.
HOW TO
Given two rational expressions, divide them.
1. Rewrite as the first rational expression multiplied by the reciprocal of the second.
2. Factor the numerators and denominators.
3. Multiply the numerators.
4. Multiply the denominators.
5. Simplify.
EXAMPLE 3
Dividing Rational Expressions
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2. Factor the numerators and denominators.
3. Multiply the numerators.
4. Multiply the denominators.
5. Simplify.
EXAMPLE 3
Dividing Rational Expressions
Divide the rational expressions and express the quotient in simplest form:
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...
Solution
TRY IT #3 Divide the rational expressions and express the quotient in simplest form:
Adding and Subtracting Rational Expressions
Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions,
we need to find a common denominator. Let’s look at an example of fraction addition.
We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same
thing when adding or subtracting rational expressions.
The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest
multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions
and multiply all of the distinct factors. For instance, if the factored denominators were and
then the LCD would be
Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the
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then the LCD would be
Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the
LCD. We would need to multiply the expression with a denominator of by and the expression with a
denominator of by
HOW TO
Given two rational expressions, add or subtract them.
1. Factor the numerator and denominator.
2. Find the LCD of the expressions.
3. Multiply the expressions by a form of 1 that changes the denominators to the LCD.
4. Add or subtract the numerators.
5. Simplify.
EXAMPLE 4
Adding Rational Expressions
Add the rational expressions:
Solution
1.6 • Rational Expressions 71
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...
First, we have to find the LCD. In this case, the LCD will be We then multiply each expression by the appropriate form
of 1 to obtain as the denominator for each fraction.
Now that the expressions have the same denominator, we simply add the numerators to find the sum.
Analysis
Multiplying by or does not change the value of the original expression because any number divided by itself is 1,
and multiplying an expression by 1 gives the original expression.
EXAMPLE 5
Subtracting Rational Expressions
Subtract the rational expressions:
Solution
Q&A Do we have to use the LCD to add or subtract rational expressions?
No. Any common denominator will work, but it is easiest to use the LCD.
TRY IT #4 Subtract the rational expressions:
Simplifying Complex Rational Expressions
A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the
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Simplifying Complex Rational Expressions
A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the
denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as
single rational expressions and dividing. The complex rational expression can be simplified by rewriting the
numerator as the fraction and combining the expressions in the denominator as We can then rewrite the
expression as a multiplication problem using the reciprocal of the denominator. We get which is equal to
HOW TO
Given a complex rational expression, simplify it.
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1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.
2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.
3. Rewrite as the numerator divided by the denominator.
4. Rewrite as multiplication.
5. Multiply.
6. Simplify.
EXAMPLE 6
Simplifying Complex Rational Expressions
Simplify: .
Solution
Begin by combining the expressions in the numerator into one expression.
Now the numerator is a single rational expression and the denominator is a single rational expression.
We can rewrite this as division, and then multiplication.
TRY IT #5 Simplify:
Q&A Can a complex rational expression always be simplified?
Yes. We can always rewrite a complex rational expression as a simplified rational expression.
MEDIA
Access these online resources for additional instruction and practice with rational expressions.
Simplify Rational Expressions (http://openstax.org/l/simpratexpress)
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MEDIA
Access these online resources for additional instruction and practice with rational expressions.
Simplify Rational Expressions (http://openstax.org/l/simpratexpress)
Multiply and Divide Rational Expressions (http://openstax.org/l/multdivratex)
Add and Subtract Rational Expressions (http://openstax.org/l/addsubratex)
Simplify a Complex Fraction (http://openstax.org/l/complexfract)
1.6 • Rational Expressions 73
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1.6 SECTION EXERCISES
Verbal
1. How can you use factoring
to simplify rational
expressions?
2. How do you use the LCD to
combine two rational
expressions?
3. Tell whether the following
statement is true or false
and explain why: You only
need to find the LCD when
adding or subtracting
rational expressions.
Algebraic
For the following exercises, simplify the rational expressions.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13.
For the following exercises, multiply the rational expressions and express the product in simplest form.
14. 15. 16.
17. 18. 19.
20. 21. 22.
23.
For the following exercises, divide the rational expressions.
24. 25. 26.
27. 28. 29.
30. 31. 32.
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For the following exercises, add and subtract the rational expressions, and then simplify.
33. 34. 35.
36. 37. 38.
39. 40. 41.
For the following exercises, simplify the rational expression.
42. 43. 44.
45. 46. 47.
48. 49. 50.
Real-World Applications
51. Brenda is placing tile on her
bathroom floor. The area of the
floor is ft2. The area
of one tile is To find
the number of tiles needed,
simplify the rational expression:
52. The area of Lijuan's yard is
ft2. A patch of
sod has an area of
ft2. Divide
the two areas and simplify
to find how many pieces of
sod Lijuan needs to cover
her yard.
53. Elroi wants to mulch his
garden. His garden is
ft2. One bag
of mulch covers
ft2. Divide the expressions
and simplify to find how
many bags of mulch Elroi
needs to mulch his garden.
Extensions
For the following exercises, perform the given operations and simplify.
54. 55. 56.
57.
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Chapter Review
Key Terms
algebraic expression constants and variables combined using addition, subtraction, multiplication, and division
associative property of addition the sum of three numbers may be grouped differently without affecting the result;
in symbols,
associative property of multiplication the product of three numbers may be grouped differently without affecting
the result; in symbols,
base in exponential notation, the expression that is being multiplied
binomial a polynomial containing two terms
coefficient any real number in a polynomial in the form
commutative property of addition two numbers may be added in either order without affecting the result; in
symbols,
commutative property of multiplication two numbers may be multiplied in any order without affecting the result; in
symbols,
constant a quantity that does not change value
degree the highest power of the variable that occurs in a polynomial
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symbols,
constant a quantity that does not change value
degree the highest power of the variable that occurs in a polynomial
difference of squares the binomial that results when a binomial is multiplied by a binomial with the same terms, but
the opposite sign
distributive property the product of a factor times a sum is the sum of the factor times each term in the sum; in
symbols,
equation a mathematical statement indicating that two expressions are equal
exponent in exponential notation, the raised number or variable that indicates how many times the base is being
multiplied
exponential notation a shorthand method of writing products of the same factor
factor by grouping a method for factoring a trinomial in the form by dividing the x term into the sum of
two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire
expression
formula an equation expressing a relationship between constant and variable quantities
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expression
formula an equation expressing a relationship between constant and variable quantities
greatest common factor the largest polynomial that divides evenly into each polynomial
identity property of addition there is a unique number, called the additive identity, 0, which, when added to a
number, results in the original number; in symbols,
identity property of multiplication there is a unique number, called the multiplicative identity, 1, which, when
multiplied by a number, results in the original number; in symbols,
index the number above the radical sign indicating the nth root
integers the set consisting of the natural numbers, their opposites, and 0:
inverse property of addition for every real number there is a unique number, called the additive inverse (or
opposite), denoted which, when added to the original number, results in the additive identity, 0; in symbols,
inverse property of multiplication for every non-zero real number there is a unique number, called the
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inverse property of multiplication for every non-zero real number there is a unique number, called the
multiplicative inverse (or reciprocal), denoted which, when multiplied by the original number, results in the
multiplicative identity, 1; in symbols,
irrational numbers the set of all numbers that are not rational; they cannot be written as either a terminating or
repeating decimal; they cannot be expressed as a fraction of two integers
leading coefficient the coefficient of the leading term
leading term the term containing the highest degree
least common denominator the smallest multiple that two denominators have in common
monomial a polynomial containing one term
natural numbers the set of counting numbers:
order of operations a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to
operations
perfect square trinomial the trinomial that results when a binomial is squared
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operations
perfect square trinomial the trinomial that results when a binomial is squared
polynomial a sum of terms each consisting of a variable raised to a nonnegative integer power
principal nth root the number with the same sign as that when raised to the nth power equals
principal square root the nonnegative square root of a number that, when multiplied by itself, equals
radical the symbol used to indicate a root
radical expression an expression containing a radical symbol
radicand the number under the radical symbol
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rational expression the quotient of two polynomial expressions
rational numbers the set of all numbers of the form where and are integers and Any rational number
may be written as a fraction or a terminating or repeating decimal.
real number line a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent
0; positive numbers lie to the right of 0 and negative numbers to the left.
real numbers the sets of rational numbers and irrational numbers taken together
scientific notation a shorthand notation for writing very large or very small numbers in the form where
and is an integer
term of a polynomial any of a polynomial in the form
trinomial a polynomial containing three terms
variable a quantity that may change value
whole numbers the set consisting of 0 plus the natural numbers:
Key Equations
Rules of Exponents
For nonzero real numbers and and integers and
Product rule
Quotient rule
Power rule
Zero exponent rule
Negative rule
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Key Equations
Rules of Exponents
For nonzero real numbers and and integers and
Product rule
Quotient rule
Power rule
Zero exponent rule
Negative rule
Power of a product rule
Power of a quotient rule
perfect square trinomial
difference of squares
difference of squares
perfect square trinomial
sum of cubes
difference of cubes
Key Concepts
1.1 Real Numbers: Algebra Essentials
• Rational numbers may be written as fractions or terminating or repeating decimals. See Example 1 and Example 2.
• Determine whether a number is rational or irrational by writing it as a decimal. See Example 3.
• The rational numbers and irrational numbers make up the set of real numbers. See Example 4. A number can be
classified as natural, whole, integer, rational, or irrational. See Example 5.
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• The order of operations is used to evaluate expressions. See Example 6.
• The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of
real numbers. These are the commutative properties, the associative properties, the distributive property, the
identity properties, and the inverse properties. See Example 7.
• Algebraic expressions are composed of constants and variables that are combined using addition, subtraction,
multiplication, and division. See Example 8. They take on a numerical value when evaluated by replacing variables
with constants. See Example 9, Example 10, and Example 12
• Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or
evaluated as any mathematical expression. See Example 11 and Example 13.
1.2 Exponents and Scientific Notation
• Products of exponential expressions with the same base can be simplified by adding exponents. See Example 1.
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1.2 Exponents and Scientific Notation
• Products of exponential expressions with the same base can be simplified by adding exponents. See Example 1.
• Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See Example
2.
• Powers of exponential expressions with the same base can be simplified by multiplying exponents. See Example 3.
• An expression with exponent zero is defined as 1. See Example 4.
• An expression with a negative exponent is defined as a reciprocal. See Example 5 and Example 6.
• The power of a product of factors is the same as the product of the powers of the same factors. See Example 7.
• The power of a quotient of factors is the same as the quotient of the powers of the same factors. See Example 8.
• The rules for exponential expressions can be combined to simplify more complicated expressions. See Example 9.
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• The rules for exponential expressions can be combined to simplify more complicated expressions. See Example 9.
• Scientific notation uses powers of 10 to simplify very large or very small numbers. See Example 10 and Example 11.
• Scientific notation may be used to simplify calculations with very large or very small numbers. See Example 12 and
Example 13.
1.3 Radicals and Rational Exponents
• The principal square root of a number is the nonnegative number that when multiplied by itself equals See
Example 1.
• If and are nonnegative, the square root of the product is equal to the product of the square roots of and
See Example 2 and Example 3.
• If and are nonnegative, the square root of the quotient is equal to the quotient of the square roots of and
See Example 4 and Example 5.
• We can add and subtract radical expressions if they have the same radicand and the same index. See Example 6 and
Example 7.
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See Example 4 and Example 5.
• We can add and subtract radical expressions if they have the same radicand and the same index. See Example 6 and
Example 7.
• Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square
root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the
denominator. See Example 8 and Example 9.
• The principal nth root of is the number with the same sign as that when raised to the nth power equals These
roots have the same properties as square roots. See Example 10.
• Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See Example 11
and Example 12.
• The properties of exponents apply to rational exponents. See Example 13.
1.4 Polynomials
• A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is
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1.4 Polynomials
• A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is
the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest
degree, and the leading coefficient is the coefficient of that term. See Example 1.
• We can add and subtract polynomials by combining like terms. See Example 2 and Example 3.
• To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in
the second. Then add the products. See Example 4.
• FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See Example 5.
• Perfect square trinomials and difference of squares are special products. See Example 6 and Example 7.
• Follow the same rules to work with polynomials containing several variables. See Example 8.
1.5 Factoring Polynomials
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• Follow the same rules to work with polynomials containing several variables. See Example 8.
1.5 Factoring Polynomials
• The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first
step in any factoring problem. See Example 1.
• Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a
sum of the second term. See Example 2.
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• Trinomials can be factored using a process called factoring by grouping. See Example 3.
• Perfect square trinomials and the difference of squares are special products and can be factored using equations.
See Example 4 and Example 5.
• The sum of cubes and the difference of cubes can be factored using equations. See Example 6 and Example 7.
• Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See Example 8.
1.6 Rational Expressions
• Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See
Example 1.
• We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See
Example 2.
• To divide rational expressions, multiply by the reciprocal of the second expression. See Example 3.
• Adding or subtracting rational expressions requires finding a common denominator. See Example 4 and Example 5.
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• Adding or subtracting rational expressions requires finding a common denominator. See Example 4 and Example 5.
• Complex rational expressions have fractions in the numerator or the denominator. These expressions can be
simplified. See Example 6.
Exercises
Review Exercises
Real Numbers: Algebra Essentials
For the following exercises, perform the given operations.
1. 2. 3.
For the following exercises, solve the equation.
4. 5.
For the following exercises, simplify the expression.
6. 7.
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive
answer.
8. 11 9. 0 10.
11.
Exponents and Scientific Notation
For the following exercises, simplify the expression.
12. 13. 14.
15. 16. 17.
18. 19. 20. Write the number in
standard notation:
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21. Write the number in
scientific notation:
16,340,000
Radicals and Rational Expressions
For the following exercises, find the principal square root.
22. 23. 24.
25. 26. 27.
28. 29. 30.
31. 32. 33.
34.
Polynomials
For the following exercises, perform the given operations and simplify.
35. 36. 37.
38. 39. 40.
41. 42. 43.
44.
Factoring Polynomials
For the following exercises, find the greatest common factor.
45. 46. 47.
For the following exercises, factor the polynomial.
48. 49. 50.
51. 52. 53.
54. 55. 56.
57. 58. 59.
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60.
Rational Expressions
For the following exercises, simplify the expression.
61. 62. 63.
64. 65. 66.
67. 68. 69.
70.
Practice Test
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive
answer.
1. 2.
For the following exercises, evaluate the expression.
3. 4. 5. Write the number in
standard notation:
6. Write the number in
scientific notation:
0.0000000212.
For the following exercises, simplify the expression.
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18.
19. 20.
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21. 22.
For the following exercises, factor the polynomial.
23. 24. 25.
26.
For the following exercises, simplify the expression.
27. 28. 29.
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From the air, a landscape of circular crop fields may seem random, but they are laid out and irrigated very precisely.
Farmers and irrigation providers combining agricultural science, engineering, and mathematics to achieve the most
productive and efficient array. (Credit: Modification of "Aerial Phot of Center Pivot Irrigations Systems (1)" by Soil
Science/flickr)
Chapter Outline
2.1 The Rectangular Coordinate Systems and Graphs
2.2 Linear Equations in One Variable
2.3 Models and Applications
2.4 Complex Numbers
2.5 Quadratic Equations
2.6 Other Types of Equations
2.7 Linear Inequalities and Absolute Value Inequalities
Introduction to Equations and Inequalities
Irrigation is a critical aspect of agriculture, which can expand the yield of farms and enable farming in areas not naturally
viable for crops. But the materials, equipment, and the water itself are expensive and complex. To be efficient and
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viable for crops. But the materials, equipment, and the water itself are expensive and complex. To be efficient and
productive, farm owners and irrigation specialists must carefully lay out the network of pipes, pumps, and related
equipment. The available land can be divided into regular portions (similar to a grid), and the different sizes of irrigation
systems and conduits can be installed within the plotted area.
2.1 The Rectangular Coordinate Systems and Graphs
Learning Objectives
In this section, you will:
Plot ordered pairs in a Cartesian coordinate system.
Graph equations by plotting points.
Graph equations with a graphing utility.
Find x-intercepts and y-intercepts.
Use the distance formula.
Use the midpoint formula.
EQUATIONS AND INEQUALITIES
2
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Figure 1
Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is
indicated by a red dot in Figure 1. Laying a rectangular coordinate grid over the map, we can see that each stop aligns
with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in
locations.
Plotting Ordered Pairs in the Cartesian Coordinate System
An old story describes how seventeenth-century philosopher/mathematician René Descartes, while sick in bed, invented
the system that has become the foundation of algebra. According to the story, Descartes was staring at a fly crawling on
the ceiling when he realized that he could describe the fly’s location in relation to the perpendicular lines formed by the
adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each
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adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each
axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just
two numbers—the displacement from the horizontal axis and the displacement from the vertical axis.
While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, it was Descartes who
introduced the components that comprise the Cartesian coordinate system, a grid system having perpendicular axes.
Descartes named the horizontal axis the x-axis and the vertical axis the y-axis.
The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane
consisting of the x-axis and the y-axis. Perpendicular to each other, the axes divide the plane into four sections. Each
section is called a quadrant; the quadrants are numbered counterclockwise as shown in Figure 2
Figure 2
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section is called a quadrant; the quadrants are numbered counterclockwise as shown in Figure 2
Figure 2
The center of the plane is the point at which the two axes cross. It is known as the origin, or point From the origin,
each axis is further divided into equal units: increasing, positive numbers to the right on the x-axis and up the y-axis;
decreasing, negative numbers to the left on the x-axis and down the y-axis. The axes extend to positive and negative
infinity as shown by the arrowheads in Figure 3.
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Figure 3
Each point in the plane is identified by its x-coordinate, or horizontal displacement from the origin, and its
y-coordinate, or vertical displacement from the origin. Together, we write them as an ordered pair indicating the
combined distance from the origin in the form An ordered pair is also known as a coordinate pair because it
consists of x- and y-coordinates. For example, we can represent the point in the plane by moving three units to
the right of the origin in the horizontal direction, and one unit down in the vertical direction. See Figure 4.
Figure 4
When dividing the axes into equally spaced increments, note that the x-axis may be considered separately from the
y-axis. In other words, while the x-axis may be divided and labeled according to consecutive integers, the y-axis may be
divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against
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