LESSON
9.2
Name
Multiplying and Dividing Rational Expressions
Class
9.2
Date
Multiplying and Dividing Rational Expressions
Essential Question: How can you multiply and divide rational expressions?
Resource Locker
Common Core Math Standards The student is expected to:
Explore
A-APR.7(+)
Use the facts you know about multiplying rational numbers to determine how to multiply rational expressions.
Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Also F-BF.1b, A-CED.4
A
Mathematical Practices Language Objective
20 5 =_ 4 ⋅_ _ 5 6 30
C
To simplify, factor the numerator and denominator. 20 = 2 ⋅ 2 ⋅ 5
Explain to a partner the steps for multiplying and dividing rational expressions.
30 = 2 ⋅ 3 ⋅ 5
D
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
Cancel common factors in the numerator and denominator to simplify the product.
2 2⋅2⋅5 = _ 5 =_ 20 = _ 4 ⋅_ _ 5 6 30 2⋅3⋅5 3
E
Based on the steps used for multiplying rational numbers, how can you multiply the rational x + 1 ______ expression ____ ⋅ 3 ? x-1 2(x + 1)
Multiply x + 1 by 3 to find the numerator of the product, and multiply x - 1 by 2(x + 1) to find the denominator. Then cancel common factors to simplify the product.
Reflect
1.
Discussion Multiplying rational expressions is similar to multiplying rational numbers. Likewise, dividing rational expressions is similar to dividing rational numbers. How could you use the steps for dividing rational numbers to divide rational expressions? When dividing rational numbers, multiply by the reciprocal of the divisor and follow the steps for multiplying rational numbers. So, when dividing rational expressions, multiply by the reciprocal of the divisor and follow the steps for multiplying rational expressions.
Module 9
PREVIEW: LESSON PERFORMANCE TASK
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HARDCOVER PAGES 317324
expressions.
Turn to these pages to find this lesson in the hardcover student edition.
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Resource Locker
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Quest Essential
© Houghto
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View the Engage section online. Discuss the photo and how the heat generated by a runner’s body could depend on height. Then preview the Lesson Performance Task.
be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
Harcour t
To find the product of rational expressions, factor each numerator and denominator, multiply the numerators and denominators, and simplify the resulting rational expression that is the product. To find the quotient of rational expressions, multiply the dividend by the reciprocal of the divisor and then follow the steps for multiplying rational expressions.
5? 4 ⋅_ How do you multiply _ 5 6 Multiply 4 by 5 to find the numerator of the product, and multiply 5 by 6 to find the denominator .
B
MP.8 Patterns
Essential Question: How can you multiply and divide rational expressions?
Relating Multiplication Concepts
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Explain 1
Multiplying Rational Expressions
EXPLORE
To multiply rational expressions, multiply the numerators to find the numerator of the product, and multiply the denominators to find the denominator. Then, simplify the product by cancelling common factors.
Relating Multiplication Concepts
Note the excluded values of the product, which are any values of the variable for which the expression is undefined. Example 1
Find the products and any excluded values.
2x 2-6x-20 3x 2 _ ⋅ __ x 2-2x-8 x 2-3x-10
INTEGRATE TECHNOLOGY
2(x + 2)(x - 5) 3x 3x 2x -6x-20 = __ _ ⋅ __ ⋅ __ x 2-2x-8 x 2-3x-10 (x + 2)(x - 4) (x + 2)(x - 5) 2
2
2
6x 2(x + 2)(x - 5) = ___ (x + 2)(x - 4)(x + 2)(x - 5) 6x 2(x + 2)(x - 5) = ___ (x + 2)(x - 4)(x + 2)(x - 5)
Students have the option of completing the Explore activity either in the book or online.
Factor the numerators and denominators. Multiply the numerators and multiply the denominators.
QUESTIONING STRATEGIES
Cancel the common factors in the numerator and denominator.
What are two different ways of 2x2 y 3y multiplying _____ · ___ ? Multiply across and 6xy 4x then simplify the result, or divide out common factors of the numerators and denominators and then multiply across. In either case, the result y will be __. 4
6x 2 = __ (x + 2)(x - 4)
Determine what values of x make each expression undefined. 3x 2 __ : The denominator is 0 when x = -2 and x = 4. x 2 - 2x - 8 2 2x - 6x - 20 : The denominator is 0 when x = -2 and x = 5. __ x 2 - 3x - 10
Excluded values: x = -2, x = 4, and x = 5
7x + 35 x 2 - 8x __ ⋅_ x+8 14(x 2 + 8x + 15)
(
)
x (x - 8) 7 x+5 7x + 35 x 2 - 8x __ ⋅ _ = __ ⋅ __ Factor the numerators and x+8 x+8 14(x 2 + 8x + 15) denominators. 14 x + 3 (x + 5)
(
x(x - 8) = __ 2(x + 3)(x + 8)
Multiply the numerators and multiply the denominators.
Cancel the common factors in the numerator and denominator.
Determine what values of x make each expression undefined. x 2 - 8x __ : The denominator is 0 when x = -3 and x = -5 . 14(x 2 + 8x + 15) 7x + 35 _ : The denominator is 0 when x = -8 . x+8 Excluded values:
EXPLAIN 1 Multiplying Rational Expressions
© Houghton Mifflin Harcourt Publishing Company
) 7x(x - 8)( x + 5 ) = ___ 14( x + 3 )(x + 5)(x + 8)
AVOID COMMON ERRORS Students sometimes confuse multiplying rational expressions with cross-multiplying. Point out that cross-multiplying takes place across an equal sign c . Tell a = __ when solving equations of the form __ b d students to use the equal sign as the cue to cross multiply. When multiplying rational expressions, multiply straight across.
x = -3, x = -5, and x = -8
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QUESTIONING STRATEGIES
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Learning Progressions Students learned how to simplify rational expressions in the previous lesson. They also know how to multiply and divide numerical fractions. Here, they combine those skills to multiply and divide rational expressions. Students apply their knowledge of factoring, as well as of multiplying polynomials, to simplify expressions involving multiplication and division of rational expressions. The concept of excluded values will carry over into later studies, for example, in excluding extraneous values in the simplification of logarithms.
16/10/14 2:09 PM
Why should you factor the numerators and the denominators before you multiply? It makes it easier to multiply because you can divide out common factors from a numerator and a denominator before multiplying.
Multiplying and Dividing Rational Expressions
440
Your Turn
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Students should recognize that multiplying
Find the products and any excluded values. 2.
x2 - 9 x-8 __ ⋅_ x 2 - 5x - 24 2x 2 - 18x
=
two rational expressions does not introduce excluded values. The excluded values of the product are the combined excluded values of the original rational expressions. Students can use this fact to help detect errors in their work.
=
(x + 3)(x - 8)
2x(x - 9)
(x + 3)(x - 3)(x - 8)
___________________
2x(x + 3)(x - 8)(x - 9)
3x =_ x+1
(x - 3) 2x(x - 9)
Excluded values: x = -1 and x = 9
Excluded values: x = -3, x = 8, x = 0, and x = 9
Explain 2
Dividing Rational Expressions
To divide rational expressions, change the division problem to a multiplication problem by multiplying by the reciprocal. Then, follow the steps for multiplying rational expressions.
Dividing Rational Expressions
Example 2
Find the quotients and any excluded values.
(x + 7) x 2 + 9x + 14 _ ÷ __ x2 x2 + x - 2 (x + 7) 2 __ (x + 7) 2 x2 + x - 2 x 2 + 9x + 14 _ ÷ __ =_ ⋅ 2 x + 9x + 14 x2 x2 x2 + x - 2 2
QUESTIONING STRATEGIES
© Houghton Mifflin Harcourt Publishing Company
Why must you exclude values of the variable that make the numerator of the divisor 0? If the numerator of a fraction is 0, then the fraction equals 0. Since division by 0 is undefined, the divisor cannot be equal to 0.
3x - 27 x ⋅_ _ x-9 x+1 3(x - 9) x = _ ⋅ _______ (x - 9) x + 1 3x(x - 9) = __ (x - 9)(x + 1)
(x + 3)(x - 3) _ x-8 ____________ ⋅
= ________
EXPLAIN 2
How is the procedure for dividing rational expressions related to multiplying rational expressions? Dividing by an expression is equivalent to multiplying by its reciprocal. Once division is converted to multiplication, you can carry out the steps for multiplying rational expressions.
3.
Multiply by the reciprocal.
(x + 7)(x + 7) __ (x + 2)(x - 1) = __ ⋅ x2 (x + 7)(x + 2)
Factor the numerators and denominators.
(x + 7)(x + 7)(x + 2)(x - 1) = ___ x 2(x + 7)(x + 2)
Multiply the numerators and multiply the denominators.
(x + 7)(x + 7)(x + 2)(x - 1) = ___ 2 x (x + 7)(x + 2)
Cancel the common factors in the numerator and denominator.
(x + 7)(x - 1) = __ x2 Determine what values of x make each expression undefined.
(x + 7) 2 _ : x
The denominator is 0 when x = 0.
x + 9x + 14 __ : The denominator is 0 when x = -2 and x = 1. x2 + x - 2 2
x2 + x - 2 __ : The denominator is 0 when x = -7 and x = -2. x 2 + 9x + 14 Excluded values: x = 0, x = -7, x = 1, and x = -2
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Peer-to-Peer Activity Have students work in pairs. Instruct each pair to create a problem involving the division of two rational expressions by working backward from the factored form of the numerators and denominators. Have them rewrite the problem, multiplying the factors in each numerator and denominator. Then have them exchange problems with another pair, and find the quotient. Have each pair compare their answer to the answer determined by the students who created the problem.
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B
9x 2 - 27x - 36 6x ÷ __ _ 3x - 30 x 2 - 10x
x 2 - 10x 6x ⋅ __ 6x ÷ __ 9x 2 - 27x - 36 = _ _ 3x - 30 3x - 30 x 2 - 10x 9x 2 - 27x - 36
(
)
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Prompt students to recognize that they can
Multiply by the reciprocal.
x x - 10 6x = __ ⋅ __ Factor the numerators and denominators. 3 x - 10 9(x + 1) x - 4
(
) 6x (
)
(
)
2 x - 10 = ___ 27 x - 10 (x + 1) x - 4
(
)
(
)
check their solutions to division problems by multiplying the quotient by the divisor and checking to see that the result is the dividend.
Multiply the numerators and multiply the denominators.
2x 2 = __ 9(x + 1)(x - 4)
Cancel the common factors in the numerator and denominator.
Determine what values of x make each expression undefined. 6x _ : 3x - 30
The denominator is 0 when x = 10 .
9x 2 - 27x - 36 : The denominator is 0 when x = 10 and x = 0 . __ x 2 - 10x x 2 - 10x __ : The denominator is 0 when x = -1 and x = 4 . 9x 2 - 27x - 36
Excluded values:
x = 0, x = 10, x = -1, and x = 4
Your Turn
Find the quotients and any excluded values. x + 11 2x + 6 _ ÷_ 4x x 2 + 2x − 3
5.
20 ÷ __ 5x 2 − 40x _ 2 x 2 − 7x x − 15x + 56
∙ _________ = ______ 4x
20 ∙ = _______ 2
x − 15x + 56 ____________
(x − 1)(x + 3) (x + 11) ____________ = _______ ∙ 4x 2(x + 3)
20 ∙ = _______
(x − 8)(x − 7) ____________
x + 11
x 2 + 2x − 3 2x + 6
x − 7x
x(x − 7)
(x + 11)(x − 1)(x + 3) = __________________ 8x(x + 3 )
© Houghton Mifflin Harcourt Publishing Company
4.
2
5x 2 − 40x
5x(x − 8)
= ______________ 2 20(x − 8)(x − 7)
5x (x − 7)(x − 8)
(x + 11)(x − 1) = _____________
4 = __ 2
Excluded values: x = 0, x = 1, and x = −3
Excluded values: x = 0, x = 7, and x = 8
8x
x
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Graphic Organizers Have students copy and complete the graphic organizer shown below, writing a worked-out example in each box. Numerical Fractions
Rational Expressions
Adding Subtracting Multiplying Dividing
Multiplying and Dividing Rational Expressions
442
Explain 3
EXPLAIN 3
Activity: Investigating Closure
A set of numbers is said to be closed, or to have closure, under a given operation if the result of the operation on any two numbers in the set is also in the set.
Activity: Investigating Closure
A
AVOID COMMON ERRORS Students may think that a single example is sufficient to prove that a set is closed. While a single counterexample is enough to prove that a set is not closed, the general result must be proven to show closure. For example, the quotient of the integer division 8 ÷ 2 = 4 is an integer, but the integers are not closed under division.
B
Recall whether the set of whole numbers, the set of integers, and the set of rational numbers are closed under each of the four basic operations.
Addition
Subtraction
Multiplication
Division
Whole Numbers
Closed
Not Closed
Closed
Not Closed
Integers
Closed
Closed
Closed
Not Closed
Rational Numbers
Closed
Closed
Closed
Closed
p(x)
r(x)
Look at the set of rational expressions. Use the rational expressions ___ and ___ where p(x), q(x) s(x) q(x), r(x) and s(x) are nonzero. Add the rational expressions. p(x) r(x) p(x)s(x) + q(x)r(x) _ + _ = q(x)s(x) q(x) s(x)
______________
C
Is the set of rational expressions closed under addition? Explain. p(x)s(x) + q(x)r(x) Yes; since q(x) and s(x) are nonzero, q(x)s(x) is nonzero. So, ______________ is again a q(x)s(x)
QUESTIONING STRATEGIES
rational expression.
How do you determine whether a set of polynomials or rational expressions is closed under a given operation? Define the members of the set. Then investigate the set to determine whether the given operation always results in a member of the set.
counter example to show that a set is not closed than to explain why a set is closed. Encourage students to use variables such as a and b to represent elements of the set, and try to determine the general result of the operation on a and b.
Subtract the rational expressions. p(x) r(x) p(x)s(x) - q(x)r(x) _ - _ = q(x) s(x) q(x)s(x)
______________
E
Is the set of rational expressions closed under subtraction? Explain. p(x)s(x) - q(x)r(x) Yes; since q(x) and s(x) are nonzero, q(x)s(x) is nonzero. So, ______________ is again a q(x)s(x)
rational expression. © Houghton Mifflin Harcourt Publishing Company
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 For most students, it will be easier to give a
D
F
Multiply the rational expressions. p(x) _ r(x) p(x)r(x) _ ∙ = q(x)s(x) q(x) s(x)
G
Is the set of rational expressions closed under multiplication? Explain. p(x)r(x) Yes; since q(x) and s(x) are nonzero, q(x)s(x) is nonzero. So, is again a rational
______
______ q(x)s(x)
expression.
H
Divide the rational expressions. p(x)s(x) p(x) r(x) _ ÷ _= q(x)r(x) q(x) s(x)
I
Is the set of rational expressions closed under division? Explain. p(x)s(x) Yes; since q(x) and r(x) are nonzero, q(x)r(x) is nonzero. So, ______ is again a rational
______
q(x)r(x)
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Reflect
6.
EXPLAIN 4
Are rational expressions most like whole numbers, integers, or rational numbers? Explain. Rational expressions are like rational numbers because both the set of rational
Multiplying and Dividing with Rational Models
expressions and the set of rational numbers are closed under all four basic operations.
Explain 4
Multiplying and Dividing with Rational Models
Models involving rational expressions can be solved using the same steps to multiply or divide rational expressions. Example 3
QUESTIONING STRATEGIES
Solve the problems using rational expressions.
How do you determine the excluded values in a real-world problem that involves dividing two rational expressions? Find the values that make each denominator 0 and that make the numerator of the divisor 0. Also, determine numbers that are not reasonable values for the independent variable in the situation.
Leonard drives 40 miles to work every day. One-fifth of his drive is on city roads, where he averages 30 miles per hour. The other part of his drive is on a highway, where d cr h + d hr c he averages 55 miles per hour. The expression ________ r cr h represents the total time spent driving, in hours. In the expression, d c represents the distance traveled on city roads, d h represents the distance traveled on the highway, r c is the average speed on city roads, and r h is the average speed on the highway. Use the expression to find the average speed of Leonard’s drive.
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Discuss with students how the rational
The total distance traveled is 40 miles. Find an expression for the average speed, r, of Leonard’s drive.
r cr h = 40 ∙ _ d cr h + d hr c 40r cr h =_ d cr h + d hr c Find the values of d c and d h . 1 (40) = 8 miles dc = _ 5 d h = 40 - 8 = 32 miles
Solve for r by substituting in the given values from the problem. dr cr h r= _ d cr h + d hr c 40 ∙ 55 ∙ 30 = __ 8 ∙ 55 + 32 ∙ 30
≈ 47 miles per hour
The average speed of Leonard’s drive is about 47 miles per hour.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ocean/ Corbis
r = Total distance traveled ÷ Total time d cr h + d hr c = 40 ÷ _ r cr h
expressions used in the example model the situation. Discuss what each numerator and denominator represents, and why a quotient of these quantities is an appropriate model.
The fuel efficiency of Tanika’s car at highway speeds is 35 miles per gallon. The - 216 ________ expression 48E represents the total gas consumed, in gallons, when Tanika drives E( E - 6 ) 36 miles on a highway and 12 miles in a town to get to her relative’s house. In the expression, E represents the fuel efficiency, in miles per gallon, of Tanika’s car at highway speeds. Use the expression to find the average rate of gas consumed on her trip.
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Multiplying and Dividing Rational Expressions
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The total distance traveled is 48 miles. Find an expression for the average rate of gas consumed, g, on Tanika’s trip.
ELABORATE
g = Total gas consumed ÷ Total distance traveled
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Call upon students to describe each step
48E - 216 ÷ =_ E(E - 6)
48
(
)
35 48 - 216 g = ___ 48 35 35 - 6
(
)(
)
1464 =_ 48,720
48E - 216 = __ 48 E (E - 6)
involved in the solution to a problem involving division of two rational expressions. Make sure they use accurate mathematical language in describing not only the division process, but also how to identify excluded values of the variable.
Solve for g by substituting in the value of E.
≈ 0.03
The value of E is 35 .
The average rate of gas consumed on Tanika’s trip is about 0.03 gallon per mile. Your Turn
7.
SUMMARIZE THE LESSON How do you divide two rational expressions? Multiply the first rational expression by the reciprocal of the second. Factor each numerator and denominator, and then multiply numerators and multiply denominators. Divide out common factors of the numerators and denominators.
The distance traveled by a car undergoing constant acceleration, a, for a time, t, is given by d = v 0t + 1 2 _ at , where v 0 is the initial velocity of the car. Two cars are side by side with the same initial velocity. One 2 car accelerates and the other car does not. Write an expression for the ratio of the distance traveled by the accelerating car to the distance traveled by the nonaccelerating car as a function of time.
Let A be the acceleration of the accelerating car. Accelerating car:
1 2 At d = v 0t + __ 2
Nonaccelerating car:
1( ) d = v t + __ 0 t 0
2
0
= v 0t
2
v 0t
Distance of nonaccelerating car
2
__1
v t + 2 At Distance of accelerating car _________________________ = ________
__1 At
2 0 = ___ + ____ v 0t v 0t
vt
2
At = 1 + ___ 2v 0
At The ratio as a function of time is 1 + ___ . 2v
© Houghton Mifflin Harcourt Publishing Company
0
Elaborate 8.
Explain how finding excluded values when dividing one rational expression by another is different from multiplying two rational expressions. When finding excluded values of a product of two rational expressions, find the values
of x for which the denominator of either expression is 0. When finding excluded values when dividing one rational expression by another, find the values of x for which the denominator of either expression or the numerator of the second expression is 0. 9.
Essential Question Check-In How is dividing rational expressions related to multiplying rational expressions? When dividing rational expressions, find the reciprocal of the divisor and change the division problem to a multiplication problem. Then follow the steps for multiplying rational expressions.
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Communicate Math Have students work in pairs. Provide each pair of students with some rational expressions to multiply or divide, written on sticky notes or index cards. Have the first student explain the steps to multiply rational expressions while the second student writes notes. Students switch roles and repeat the process for a division problem, highlighting the additional step of using the reciprocal.
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EVALUATE
Evaluate: Homework and Practice 1.
• Online Homework • Hints and Help • Extra Practice
Explain how to multiply the rational expressions.
2 - 3x +4 x - 3 ⋅ x_ _ _ 2 x 2 - 2x Multiply x - 3 by x 2 - 3x + 4 to get the numerator of the product. Multiply 2 by x 2 - 2x to get the denominator of the product. Then, simplify by cancelling common factors in the numerator and the denominator.
Find the products and any excluded values. x x-2 ⋅_ 2. _ 3x - 6 x + 9
ASSIGNMENT GUIDE 3.
x-2 x ⋅_ =_ 3(x - 2) x + 9
x(x - 2) = __ 3(x - 2)(x + 9) x =_ 3(x + 9)
x - 2x - 15 ⋅ __ 3 __ 10x + 30 x 2 - 3x - 10 2
(x - 5)(x + 3) 3 = __ ⋅ __ 10(x + 3) (x + 2)(x - 5) 3(x - 5)(x + 3) = ___ 10(x + 3)(x + 2)(x - 5)
Concepts and Skills
Practice
5x(x + 5) 4x =_⋅_ 2 x+5
Explore Relating Multiplication Concepts
Exercise 1
20x 2(x + 5) = __ 2(x + 5)
Example 1 Multiplying Rational Expressions
Exercises 2–7
= 10x 2
Example 2 Dividing Rational Expressions
Exercises 8–13
Example 3 Activity: Investigating Closure
Exercises 14–17
Example 4 Multiplying and Dividing with Rational Models
Exercises 18–20
Excluded value: x = -5
Excluded values: x = 2 and x = -9
4.
5x 2 + 25x _ _ ⋅ 4x x+5 2
5.
x2 - 1 x2 __ ⋅_ x 2 + 5x + 4 x 2 - x
(x - 1)(x + 1) x2 = __ ⋅ _ (x + 4)(x + 1) x(x - 1) (x - 1)(x + 1)x 2 = ___ x(x + 4)(x + 1)(x - 1)
x =_ x+ 4
and x = 5
x = 0, and x = 1
Excluded values: x = -3, x = -2,
Module 9
Excluded values: x = -4, x = -1,
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Practices
1
1 Recall of Information
MP.6 Precision
2–13
1 Recall of Information
MP.2 Reasoning
14–17
2 Skills/Concepts
MP.6 Precision
18–20
2 Skills/Concepts
MP.4 Modeling
21–22
3 Strategic Thinking
MP.2 Reasoning
2 Skills/Concepts
MP.4 Modeling
23
When multiplying rational expressions, students may divide out by common factors and then, erroneously, cross-multiply instead of multiplying straight across. Remind them that cross-multiplying is used to solve equations, and that when multiplying two rational expressions, they must multiply straight across.
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AVOID COMMON ERRORS
© Houghton Mifflin Harcourt Publishing Company
3 =_ 10(x + 2)
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6.
AVOID COMMON ERRORS
2 x 2 + 14x + 33 _ 8x - 56 __ ⋅ x - 3x ⋅ __ 4x x + 3 x 2 + 4x - 77
7.
3 9x 2 (x + 6)(x - 6) = _ ⋅ __ ⋅ _ x-6 4x(x + 6) 3(x - 2)
(x + 11)(x + 3) x(x - 3) 8(x - 7) = __ ⋅ _ ⋅ __ 4x x+3 (x + 11)(x - 7)
When identifying excluded values for quotients of rational expressions, students may consider values that cause the denominators to be zero, but they may forget to consider values that cause the divisor itself x 2 - 36 will have to be 0. For example, the divisor ______ x 2 - 4x a value of 0 when x = 6 or x = -6, so these values must also be excluded values.
9x 2 ⋅ _ x 2 - 36 ⋅ _ 3 _ x - 6 3x - 6 4x 2 + 24x
27x 2 (x + 6)(x - 6) = ___ 12x(x - 6)(x - 2)(x + 6)
8x(x + 11)(x + 3)(x - 3)(x - 7) = ___ 4x(x + 3)(x + 11)(x - 7) = 2(x - 3)
9x =_ 4(x - 2)
Excluded values: x = 0, x = -3, x = -11, and
Excluded values: x = 6, x = 2, x = 0 and x = -6
x=7 Find the quotients and any excluded values. 20x + 40 5x 2 + 10x ÷_ 8. _ x 2 + 2x + 1 x2 - 1
9.
x 2 - 9x + 18 x 2 - 36 __ ÷_ x 2 + 9x + 18 x2 - 9
=
5x + 10x _ x -1 __ ⋅
=
x - 9x + 18 _ x -9 __ ⋅
=
5x(x + 2) (x + 1)(x - 1) __ ⋅ __
=
(x + 3)(x - 3) (x - 6)(x - 3) __ __ ⋅
5x(x + 2)(x + 1)(x - 1) ___
=
x(x - 1) _
=
= =
2
x 2 + 2x + 1
2
20x + 40
20(x + 2)
(x + 1)(x + 1)
20 (x + 1)(x + 1)(x + 2)
4(x + 1)
2
2
x 2 + 9x + 18
x 2 - 36
(x + 6)(x + 3) (x + 6)(x - 6) (x - 6)(x - 3)(x + 3)(x - 3) ___
(x + 6)(x + 3)(x + 6)(x - 6)
(x - 3) _ 2
(x + 6)2
Excluded values: x = ±6, x = ±3
Excluded values: x = 1, x = -1, and
© Houghton Mifflin Harcourt Publishing Company
x = -2 -x 2 + x + 20 x+4 10. __ ÷_ 2x - 14 5x 2 - 25x
x+3 x 2 - 25 11. __ ÷_ x-5 x 2 + 8x + 15
-x + x + 20 _ 2x - 14 __ ⋅ 2
=
5x 2 - 25x
x+4
-(x + 4)(x - 5) 2(x - 7) = __ ⋅ _ 5x(x - 5)
x+4
-2(x + 4)(x - 5)(x - 7) = ___ 5x(x - 5)(x + 4)
_
Lesson 9.2
x+3 x-5 __ ⋅ __
x 2 + 8x + 15
x 2 - 25
(x + 5)(x + 3) (x + 5)(x - 5)
(x + 3)(x - 5) ___
(x + 5)(x + 3)(x + 5)(x - 5)
1 _
(x + 5) 2
Excluded values: x = -5, x = -3, and
x = -4
447
=
=
Excluded values: x = 0, x = 5, x = 7 and
A2_MNLESE385894_U4M09L2 447
x+3 x-5 __ ⋅_
=
-2(x - 7) = 5x
Module 9
=
x=5 447
Lesson 2
7/7/14 9:15 AM
x 2 - 10x + 9 x 2 - 7x - 18 12. __ ÷ __ 3x x 2 + 2x
=
x + 2x x - 10x + 9 __ __ ⋅
=
x(x + 2) (x - 1)(x - 9) __ __ ⋅
2
8x + 32 x 2 - 6x 13. __ ÷ __ x 2 + 8x + 16 x 2 - 2x - 24
2
3x
3x
x - 7x - 18 2
(x + 2)(x - 9)
x(x - 1)(x - 9)(x + 2) = ___ 3x(x + 2)(x - 9)
=
=
8x + 32 x - 2x - 24 __ ⋅ __
=
8(x + 4) (x + 4)(x - 6) __ ⋅ __
=
x-1 _
SMALL GROUP ACTIVITY Have students work in small groups to make a poster showing how to divide two rational expressions. Give each group a different problem, each consisting of polynomials that require several different factoring strategies. Then have each group present its poster to the rest of the class, explaining each step.
2
x 2 + 8x + 16
x 2 - 6x
(x + 4)(x + 4)
x(x - 6)
8(x + 4)(x + 4)(x - 6) ___ x(x + 4)(x + 4)(x - 6)
_
8 = x
3
Excluded values: x = 0, x = -2, and
Excluded values: x = 0, x = -4, and x = 6
x=9
1 1 Let p(x) = ____ and q(x) = ____ . Perform the operation, and show x+1 x-1 that it results in another rational expression.
14. p(x) + q(x)
2x __________ ; the numerator and denominator are polynomials, so it is
(x + 1)(x - 1)
a rational expression.
15. p(x) - q(x)
-2 __________ ; the numerator and denominator are polynomials, so it is
(x + 1)(x - 1)
a rational expression.
© Houghton Mifflin Harcourt Publishing Company
16. p(x) ⋅ q(x)
1 __________ ; the numerator and denominator are polynomials, so it is
(x + 1)(x - 1)
a rational expression. 17. p(x) ÷ q(x)
x-1 ____ ; the numerator and denominator are polynomials, so it is x+1
a rational expression.
Module 9
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448
Lesson 2
6/15/15 11:31 AM
Multiplying and Dividing Rational Expressions
448
18. The distance a race car travels is given by the equation d = v 0t + __12 at 2, where v 0 is the initial speed of the race car, a is the acceleration, and t is the time travelled. Near the beginning of a race, the driver accelerates for 9 seconds at a rate of 4 m/s 2. The driver’s initial speed was 75 m/s. Find the driver’s average speed during the acceleration.
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Students can use a graphing calculator to
The average speed is equal to the
compare the graph of the function defined by the original product or quotient with the graph of the function defined by the final simplified expression. If the expressions are equivalent, the graphs should be identical.
distance traveled divided by the time. d r= t 1 t v 0 + at 2 = t 1 = v 0 + at 2
_
( _ ) __ _
Substitute the known values into the equation to find r. 1 r = v 0 + at 2 1 = 75 + (4)(9) 2
_
_
= 93
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©David Madison/Corbis
The average speed during the acceleration is 93 meters per second. 19. Julianna is designing a circular track that will consist of three concentric rings. The radius of the middle ring is 6 meters greater than that of the inner ring and 6 meters less than that of the outer ring. Find an expression for the ratio of the length of the outer ring to the length of the middle ring and another for the ratio of the length of the outer ring to length of the inner ring. If the radius of the inner ring is set at 90 meters, how many times longer is the outer ring than the middle ring and the inner ring?
Length of inner ring: 2πr Length of middle track: 2π(r + 6)
Length of outer ring: 2π(r + 12)
2π(r + 12) Length of outer ring __ = __ r + 12 =_ r+6
2π(r + 12) Length of outer ring __ = __ =
A2_MNLESE385894_U4M09L2 449
Lesson 9.2
2πr
Length of inner ring
Module 9
449
2π(r + 6)
Length of middle ring
r + 12 _ r
Substitute 90 for r. 90 + 12 _ 102 _ = = 1.0625 90 + 6
96
90
90
90 + 12 _ 102 _ = ≈ 1.13 The outer ring is 1.0625 times longer than the middle ring and about 1.13 times longer than the inner ring.
449
Lesson 2
7/7/14 9:20 AM
MODELING
20. Geometry Find a rational expression for the ratio of the surface area of a cylinder to the volume of a cylinder. Then find the ratio when the radius is 3 inches and the height is 10 inches.
Surface Area = 2πr 2 + 2πrh
Substitute 3 for r and 10 for h. 2 (3 + 10) _______
Volume = πr 2h
(3)(10)
2πr + 2πrh Surface Area _________ = ________ 2
26 13 = __ = __ 15 30
The ratio of the cylinder’s surface area
πr 2h
Volume
When working with rational expressions that represent real-world situations, students should recognize that not only must they consider excluded values that are based on the algebraic nature of the rational expressions, but they also need to consider values that must be excluded due to the limitations on the domain in the given situation.
2πr(r + h) = _______ 2
to its volume is 13:15.
πr h
2(r + h) = ______ rh
H.O.T. Focus on Higher Order Thinking
21. Explain the Error Maria finds an expression equivalent to 6x 2 - 150 . x 2 - 4x - 45 ÷ _ __ x 2 - 5x 3x - 15 Her work is shown. Find and correct Maria’s mistake.
CONNECT VOCABULARY
(x - 9)(x + 5) 6(x + 5)(x - 5) x 2 - 4x - 45 6x 2 - 150 __ = __ ÷ __ ÷_ 3x - 15 x 2 - 5x ( ) 3 x-5 x(x - 5)
Have students complete a vocabulary chart using rational numbers and rational expressions, with examples of both fractions and rational expressions. Include the terms used in this lesson: numerator, denominator, factor, reciprocal.
6(x - 9)(x + 5)(x + 5)(x - 5) = ___ 3x(x - 5)(x - 5) 2 2(x - 9)(x + 5) = __ ( ) x x-5
Maria did not multiply by the reciprocal. 2 2 2 x________ - 4x - 45 x 2 - 150 - 4x - 45 _______ ÷ 6_______ = x________ ⋅ x 2 - 5x 3x - 15 3x - 15 x2 - 5x 6x - 150
(x - 9)(x + 5) x(x - 5) = _________ ⋅ __________ 3(x - 5) 6(x + 5)(x - 5)
=
x(x - 9)(x + 5)(x - 5) ________________
18x(x - 5)(x + 5)(x - 5)
x(x - 9) 18(x - 5)
© Houghton Mifflin Harcourt Publishing Company
= _______
22. Critical Thinking Multiply the expressions. What do you notice about the resulting expression? 3x + 18 3 x x - 4x + _ )( __ - _) (_ x-4 8 8x - 32 x + 2x - 24 3
2
2
(
x(x + 2)(x- 2) 3 + __________ = ____ x-4 8(x + 2)(x - 2)
(
)(
)
3 3 x ____ x +_ -_ = ____ x-4 8 x-4 8
(
3 = ____ x-4
) - (_8x) 2
)(
)
3(x + 6) x ___________ -_ 8 (x - 4)(x + 6)
2
3 x _ 3 = _____ ∙ ____ -_ ∙x x-4 x-4 8 8 x 9 _ = ______ 2 64 2
(x - 4)
The expression is the difference of two squares. Module 9
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450
Lesson 2
16/10/14 1:59 PM
Multiplying and Dividing Rational Expressions
450
23. Multi-Step Jordan is making a garden with an area of x 2 + 13x + 30 square feet and a length of x + 3 feet.
JOURNAL Have students compare and contrast the method they have learned for multiplying rational expressions with the method they have learned for adding rational expressions.
a. Find an expression for the width of Jordan’s garden. + 13x + 30 ____ (x 2 + 13x + 30) ÷ (x + 3) = x_________ ⋅ x +1 3 1 2
(x + 10)(x + 3) ____ 1 ⋅ x+ = __________ 1 3 (x + 10)(x + 3) = __________ x+3
= x + 10 Jordan’s garden is x + 10 feet wide. b. If Karl makes a garden with an area of 3x 2 + 48x + 180 square feet and a length of x + 6, how many times larger is the width of Jon’s garden than Jordan’s? x + 48x + 180 ____ (3x 2 + 48x + 180) ÷ (x + 6) = 3___________ ⋅ x +1 6 1 2
3(x + 10)(x + 6) ____ 1 ⋅ x+ = ___________ 1 6 3(x + 10)(x + 6) = ___________ x+6 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Shannon Fagan/Corbis
= 3(x + 10) Karl’s garden is 3 times wider than Jordan’s garden.
c. If x is equal to 4, what are the dimensions of both Jordan’s and Karl’s gardens?
Jordan’s garden: Length: 4 + 3 = 7 feet Width: 4 + 10 = 14 feet Karl's garden: Length: 4 + 6 = 10 feet
Width: 3(4 + 10) = 42 feet
Module 9
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451
Lesson 9.2
451
Lesson 2
7/7/14 9:31 AM
Lesson Performance Task
AVOID COMMON ERRORS
Who has the advantage, taller or shorter runners? Almost all of the energy generated by a long-distance runner is released in the form of heat. For a runner with height H and speed V, the rate h g of heat generated and the rate h r of heat released can be modeled by h g = k 1H 3V 2 and h r = k 2H 2, k 1 and k 2 being constants. So, how does a runner’s height affect the amount of heat she releases as she increases her speed?
Students may think that the amount of heat released by the runner is independent of speed because h r = k 2 H2, which is independent of V. Ask students where the heat comes from before it is released. generated by runner Then ask what the expression is for the heat generated, and ask whether it depends on V. h g = k 1 H 3V 2 ; yes Then have students write the equation for the situation that occurs when the amount of heat released is equal to the amount of heat generated.
First, set up the ratio for the amount of heat generated by the runner to the amount of heat released by dividing the value h g by h r. h kHV __ = _____ g
hr
1
3
2
k 2H 2
Next, simplify the ratio. h k HV __ = ____ g
1
2
k2 hg is equal to 1, the amount of heat released is the same as When __ hr hr
the amount of heat generated. You can use this condition as a way to determine the relationship of height to speed. Setting the ratio equal to 1,
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Have students draw a graph for the relation
isolate speed on one side of the equation. k ___ = V2 2
k 1H
Since k 1 and k 2 are constants, you see that as a runner’s height increases, the speed required to maintain the balance of heat generated to heat
they obtained between height and speed. Ask them what information they need to draw the exact graph for the relation. the constants k 1 and k 2 Then, have students discuss whether this graph helps them determine an ideal height for a runner.
released gets smaller. Therefore, a shorter runner can run at a higher speed and not lose as much heat as a taller runner does.
© Houghton Mifflin Harcourt Publishing Company
Module 9
452
Lesson 2
EXTENSION ACTIVITY A2_MNLESE385894_U4M09L2 452
Ask students to rework the problem, this time with the heat generated modeled by h g = k 1 H 3V . Ask them to describe the relation between speed and height, and to tell how that relation differs from the answer they calculated in the Performance Task. Ask them whether this model gives shorter runners a greater or lesser advantage, compared to the model in the Performance Task. Shorter runners still have an advantage over taller runners, but it is not as great.
7/7/14 9:33 AM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Multiplying and Dividing Rational Expressions
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