Algebra 1 my hrw lesson 17 module 18.3 PDF

Preview

Summary

It teaches you about adding and subtracting polynomials, and monomials s in Algebra 1 this year 2021/2022. Wish you the best of luck my fellow students, just pull through and never give up....

Description

18.1 Multiplying 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.



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

? ×

= x2 © Houghton Mif f lin Harcourt Publishing Company

? = -x2 =x = -x =1

= -1

Now fill in the table. ×

The simplified expression for 2(x + 1) = Module 18

?

x+ 663

?

. Lesson 1



Use algebra tiles to model 2x(x - 3). Then write the expression.

×

The simplified expression for 2x(x - 3) =

?

x2 -

?

x.

Reflect

1.

Discussion How do the tiles illustrate the idea of 2x geometrically?

2.

Discussion How does the grid illustrate the Distributive Property?

Explain 1

Multiplying Monomials

When multiplying monomials, variables with exponents may need to be multiplied. Recall the Product of Powers Property, which states that am ∙ a n = a(m + n ). Example 1



Find each product.

(6x3)(-4x 4 )

 (5xy )(7xy)

(6x3)(-4x 4 )

(5xy 2)(7xy)

= (6 ∙ -4)(x 3 ∙ x4)

= 5· 7

2

( )(x · x )( y ·y) = (5 · 7 )( x )(y )

= (6 ∙ -4)(x 3 + 4) = -24x

2

1 +  1

7

= 35 x 

2

y 

2 +1

3

© Houghton Mif f lin Harcourt Publishing Company

Reflect

3.

In the Product of Powers Property, do the bases need to be the same or can they be different?

Your Turn

4.

(18y2x 3z)(3x8y 6z 4)

Module 18

664

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



Find each product.

3x(3x 2 + 6x - 5) 3x(3x 2 + 6x - 5)

Distribute and simplify.

= 3x(3x2 ) + 3x (6x) + 3x (-5 ) = 9x 1 + 2 + 18x 1 + 1 - 15x 1 = 9x 3 + 18x2 - 15x



2xy(5x2y + 3xy 2 + 7xy) 2xy(5x 2y + 3xy 2 + 7xy) = 2xy(5x 2y ) + 2xy

(

Distribute and simplify.

3xy 2

= 10x 1 + 2y 1 + 1 + 6 x 1 + = 10x 3y2 + 6 x

2

y

3

1

) + 2xy (

)

7xy

 y 1 + 2 + 14 x 1 + 1

+ 14 x

2

y1 +

1

y2

Reflect

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.

2a2 (5b2 + 3ab + 6a + 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.

Module 18

665

Lesson 1

Analyze Information Identify the important information

3

.

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 = length · width · height . Use this formula and the given information to write and solve an equation.

Solve Build the equation. Since the length and width are equal, let s represent these measurements. The

(

volume will be V = (s · s)

s+6

)= s ( 2

s+6

s

s 3 + 6s 2

11

2057

12

2592

13

3211

14

3920

)= s

3

+

6

s

2

.

Justify and Evaluate 3211 is closer to 3500 than any of the other results, so the length of the fish tank to the nearest whole inch should be 13 inches.

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 in2. To the nearest whole inch, what should the dimensions of the paper be?

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?

Module 18

666

Lesson 1

© Houghton Mif f lin Harcourt Publishing Company

Your Turn

Evaluate: Homework and Practice Find each product. 1.

(3x)(2x 2)

2.

(19x5)(8x3 )

3.

(6x 7)(3x 3)

4.

(3x2)(2x 3)

5.

7xy (3x 2y3)

6.

(6xyz 4)(5xy3)

7.

(8xy3) (4y 4z2)

8.

(11xy)(x 3y2)

9.

(x 2 + x) (x 3)

10. (x3 + 2x 2)(x 4)

11. (x2 + 2x + 5)(x 3)

12. (x4 + 3x 3 + 2x 2 + 11x + 4)(x 2)

13.

(x3 + 2y 2 + 3xy)(4x 2y)

14. (2x 3 + 5y)(3xy )

15.

(x4 + 3x 3y + 3xy3)( 6xy2)

16. (x4 + 3x 3y 2 + 4x2y + 8xy + 12x)(11x 2y3)

Write a polynomial equation for each situation and then solve the problem. 1 . Design A bedroom has a length of x + 3 feet and a width of x feet. Find the area 17 when x = 10. 1 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 in2. To the nearest whole inch, what are the dimensions of the television?

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 ft3, what should the length be? Round to the nearest foot. 20. State whether each expression is or can be written as a monomial. a. x 3 b. a 2 + 2ab + b c c. x 3 + 4x3 3

d. y 2

x

e. xyz + txy + tyz + txz

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

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 2x and x3 + 5x 2 + 1 is x 6 + 5x4 + x2. Explain the error that Sandy made. 24. Communicate Mathematical Ideas What is the lowest degree that a polynomial can have? Explain.

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 = 2 h + 1 © Houghton Mif f lin Harcourt Publishing Company

Module 18

668

Lesson 1...