Download 18.1 Multiplying Polynomial Expressions by Monomials

Name Class Date 18.1Multiplying Polynomial
Expressions by Monomials
Essential Question: How can you multiply polynomials by monomials?
Resource
Locker
Explore Modeling Polynomial Multiplication
Algebra tiles can be used to model the multiplication of a polynomial by a monomial.
Rules
1. The first factor goes on the left side of the grid, the second factor on
the top.
2. Fill in the grid with tiles that have the same height as tiles on the left
and the same length as tiles on the top.
3. Follow the key. The product of two tiles of the same color is positive;
the product of two tiles of different colors is negative.
A
Use algebra tiles to find 2​(x + 1)​. Then recount the tiles in the grid and write the
expression.
First, fill in the factors.
Key
© Houghton Mifflin Harcourt Publishing Company
×
= x2
= -x2
=x
= -x
=1
= -1
Now fill in the table.
×
The simplified expression for 2​(x + 1)​ =
Module 18
x+
847
.
Lesson 1

Use algebra tiles to model 2x(x - 3). Then write the expression.
×
The simplified expression for
x2 -
2x(x - 3) =
x.
Reflect
1.
Discussion How do the tiles illustrate the idea of x 2 geometrically?
2.
Discussion How does the grid illustrate the Distributive Property?
Multiplying Monomials
Explain 1
When multiplying monomials, variables with exponents may need to be multiplied. Recall the Product of Powers
Property, which states that a m ∙ a n = a (m + n).
Example 1
A
Find each product.
 (5xy )(7xy)
(6x 3)(-4x 4)
2
(6x 3)(-4x 4)
( )(x · )(
= (5 ·
)(x )(y
= (6 ∙ -4)(x 3 ∙ x 4)
= (6 ∙ -4)(x
= -24x
= 5·
)
3+4
1 + 7
=
x y )
© Houghton Mifflin Harcourt Publishing Company
(5xy 2)(7xy)
·y
)
+1
Reflect
3.
In the Product of Powers Property, do the bases need to be the same or can they be different?
Your Turn
4.
(18y 2x 3z)(3x 8y 6z 4)
Module 18
848
Lesson 1
Explain 2
Multiplying a Polynomial by a Monomial
Remember that the Distributive Property states that multiplying a term by a sum is the same thing as multiplying the
term by each part of the sum then adding the results.
Example 2
A
Find each product.
3x(3x 2 + 6x - 5)
3x(3x 2 + 6x - 5)
Distribute and simplify.
= 3x(3x 2) + 3x(6x) + 3x (-5)
= 9x 1 + 2 + 18x 1 + 1 - 15x 1
= 9x 3 + 18x 2 - 15x

2xy(5x 2y + 3xy 2 + 7xy)
2xy(5x 2y + 3xy 2 + 7xy)
= 2xy(5x 2y) + 2xy
= 10x 1 + 2y 1 + 1 +
= 10x 3y 2 +
x
(
) + 2xy (
y 1 +
x1 +
y
+
+
x
x1 +
)
Distribute and simplify.
y1 +
y
Reflect
© Houghton Mifflin Harcourt Publishing Company
5.
Is the product of a monomial and a polynomial always a polynomial? Explain. If so, how many terms does
it have?
Your Turn
6.
2a 2(5b 2 + 3ab + 6a + 1)
Module 18
849
Lesson 1
Explain 3
Multiplying a Polynomial by a Monomial
to Solve a Real-World Problem
Knowing how to multiply polynomials and monomials is useful when solving real-world problems.
Example 3
Write a polynomial equation and solve the problem.
Design Harry is building a fish tank that is a square prism.
He wants the height of the tank to be 6 inches longer than
the length and width. If he needs the volume to be
as close as possible to 3500 in 3, what should be the length
of the tank? Round to the nearest inch.
Analyze Information
Identify the important information
• Since the bases are squares, the length and width are
• The height of the tank is
.
more inches than the length.
in 3.
• The total volume of the model should be as close as possible to
Formulate a Plan
Since the desired volume of the model is given, the volume formula should be used
to find the answer. The volume formula for a square prism is
V=
and solve an equation.
. Use this formula and the given information to write
Solve
volume will be V = (s · s)
(
s
)=s (
s
+
)=s
+
s
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jean
Michel Labat/Mary Evans Picture Library/age fotostock
Build the equation.
Since the length and width are equal, let s represent these measurements. The
.
s
11
12
13
14
Justify and Evaluate
is closer to
than any of the other results, so the length of the
fish tank to the nearest whole inch should be
Module 18
inches.
850
Lesson 1
Your Turn
7.
Engineering Diane needs a piece of paper whose length is 4 more inches than the width, and the area is
as close as possible to 50 in 2. To the nearest whole inch, what should the dimensions of the paper be?
x
x 2 + 4x
Elaborate
8.
What is the power if a monomial is multiplied by a constant?
9.
Essential Question Check-In What properties and rules are used to multiply a multi-term polynomial
by a monomial?
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
© Houghton Mifflin Harcourt Publishing Company
Find each product.
1.
(3x)(2x 2)
2.
(19x 5)(8x 3)
3.
(6x 7)(3x 3)
4.
(3x 2)(2x 3)
5.
7xy(3x 2y 3)
6.
(6xyz 4)(5xy 3)
7.
(8xy 3)(4y 4z 2)
8.
(11xy)(x 3y 2)
9.
(x 2 + x)(x 3)
Module 18
851
Lesson 1
10.
(x 3 + 2x 2)(x 4)
11.
(x 2 + 2x + 5)(x 3)
12.
(x 4 + 3x 3 + 2x 2 + 11x + 4)(x 2)
13.
(x 3 + 2y 2 + 3xy)(4x 2y)
14.
(2x 3 + 5y)(3xy)
15.
(x 4 + 3x 3y + 3xy 3)(6xy 2)
16.
(x 4 + 3x 3y 2 + 4x 2y + 8xy + 12x)(11x 2y 3)
Write a polynomial equation for each situation and then solve the problem.
17. Design A bedroom has a length of x + 3 feet and a width of x feet. Find the area
when x = 10.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Peter
Cade/Getty Images
18. Engineering A flat-screen television has
a length that is 1 more inch than its width.
The area of the television’s screen is 1500 in 2.
To the nearest whole inch, what are the
dimensions of the television?
x
Module 18
x2 + x
852
Lesson 1
19. Construction Zach is building a new shed shaped like a square prism. He wants the
height of the shed to be 2 feet less than the length and width. If he needs the volume to
be as close as possible to 3174 f​t 3​ ​, what should the length be? Round to the nearest foot.
​x ​3​
​x ​2​-2​x ​2​
20. State whether each polynomial is also a monomial.
a. x​  ​3​
Yes
No
Yes
No
c. ​x ​ ​ + 4​x ​ ​
Yes
No
d. ​y ​ ​
Yes
No
e. xyz + txy + tyz + txz
Yes
No
b. ​a ​ ​ + 2​a ​ ​+ ​b ​​
b
2
3
c
3
3
​2 ​​x ​ ​​
21. Draw the algebra tiles that model the factors in the multiplication shown. Then
determine the simplified product.
×
H.O.T. Focus on Higher Order Thinking
© Houghton Mifflin Harcourt Publishing Company
22. Critical Thinking When finding the product of a monomial and a binomial,
how is the degree of the product related to the degree of the monomial and the
degree of the binomial?
23. Explain the Error Sandy says that the product of x​  ​2​and x​  ​3​ + 5​x 2​ ​ + 1
is ​x 6​ ​ + 5​x 4​ ​ + ​x 2​ ​. Explain the error that Sandy made.
24. Communicate Mathematical Ideas What is the lowest degree that a polynomial
can have? Explain.
Module 18
853
Lesson 1
Lesson Performance Task
A craftsman is making a dulcimer with the same dimensions as the one shown. The surface
shown requires a special, more durable type of finish. Write a polynomial that represents the
area to be finished on the dulcimer shown.
b2 = h + 1
h
b1 = 2h + 1
© Houghton Mifflin Harcourt Publishing Company
Module 18
854
Lesson 1