340 Chapter 7 Polynomial Equations and Factoring

Multiplying Polynomials7.3

How can you multiply two binomials?

Work with a partner. Six different algebra tiles are shown below.

Write the product of the two binomials shown by the algebra tiles.

a. (x + 3)(x − 2) =

b. (2x − 1)(2x + 1) =

c. (x + 2)(2x − 1) = d. (− x − 2)(x − 3) =

ACTIVITY: Multiplying Binomials Using Algebra Tiles11

1 1 x x x2 x2

COMMON CORE

Polynomials In this lesson, you will● multiply binomials using

the Distributive Property, a table, or the FOIL method.

● multiply binomials and trinomials.

Learning StandardA.APR.1

Section 7.3 Multiplying Polynomials 341

Work with a partner. Write each product. Explain your reasoning.

a. b.

c. d.

e. f.

g.

h.

i.

j.

ACTIVITY: Multiplying Monomials Using Algebra Tiles22

Use what you learned about multiplying binomials to complete Exercises 3 and 4 on page 345.

4. IN YOUR OWN WORDS How can you multiply two binomials? Use the results of Activity 3 to summarize a procedure for multiplying binomials without using algebra tiles.

5. Find two binomials with the given product.

a. x 2 − 3x + 2 b. x 2 − 4x + 4

Use algebra tiles to fi nd each product.

a. (2x − 2)(2x + 1) b. (4x + 3)(x − 2)

c. (− x + 2)(2x + 2) d. (2x − 3)(x + 4)

e. (3x + 2)(− x − 1) f. (2x + 1)(− 3x + 2)

g. (x − 2)2 h. (2x − 3)2

ACTIVITY: Multiplying Binomials Using Algebra Tiles33

Use a DiagramHow can you represent the product of polynomials using diagrams?

Math Practice

342 Chapter 7 Polynomial Equations and Factoring

Lesson7.3Lesson Tutorials

Key VocabularyFOIL Method, p. 343

In Section 1.2, you used the Distributive Property to multiply a binomial by a monomial. You can also use the Distributive Property to multiply two binomials.

Find each product.

a. (x + 2)(x + 5)

Use the horizontal method.

(x + 2)(x + 5) = x(x + 5) + 2(x + 5) Distribute (x + 5) to each term of (x + 2).

= x(x) + x(5) + 2(x) + 2(5) Distributive Property

= x 2 + 5x + 2x + 10 Multiply.

= x 2 + 7x + 10 Combine like terms.

b. (x + 3)(x − 4)

Use the vertical method.

x + 3 × x − 4 Align like terms vertically.

− 4x − 12 Distributive Property

x2 + 3x Distributive Property

x 2 − x − 12 Combine like terms.

The product is x 2 − x − 12.

EXAMPLE Multiplying Binomials Using the Distributive Property11

Exercises 5–13and 16–21

Find (2x − 3)(x + 5).

Step 1: Write each binomial as a sum of terms.

(2x − 3)(x + 5) = [2x + (− 3)](x + 5)

Step 2: Make a table of products.

The product is 2x 2 − 3x + 10x − 15, or 2x 2 + 7x − 15.

Use the Distributive Property to fi nd the product.

1. ( y + 4)( y + 1) 2. (z − 2)(z + 6)

Use a table to fi nd the product.

3. ( p + 3)( p − 8) 4. (r − 5)(2r − 1)

EXAMPLE Multiplying Binomials Using a Table22

Multiply − 4(x + 3).

Multiply x(x + 3).

2x −3

x 2x 2 − 3x

5 10x − 15

Section 7.3 Multiplying Polynomials 343

The FOIL Method is a shortcut for multiplying two binomials.

FOIL Method

To multiply two binomials using the FOIL Method, fi nd the sum of the products of the

First terms, (x + 1)(x + 2) x(x) = x 2

Outer terms, (x + 1)(x + 2) x(2) = 2x

Inner terms, and (x + 1)(x + 2) 1(x) = x

Last terms. (x + 1)(x + 2) 1(2) = 2

(x + 1)(x + 2) = x 2 + 2x + x + 2 = x2 + 3x + 2

Find each product.

a. (x − 3)(x − 6)

First Outer Inner Last

(x − 3)(x − 6) = x(x) + x(− 6) + (− 3)(x) + (− 3)(− 6) Use the FOIL Method.

= x 2 + (− 6x) + (− 3x) + 18 Multiply.

= x 2 − 9x + 18 Combine like terms.

b. (2x + 1)(3x − 5)

First Outer Inner Last

(2x + 1)(3x − 5) = 2x(3x) + 2x(− 5) + 1(3x) + 1(− 5) Use the FOIL Method.

= 6x 2 + (− 10x) + 3x + (− 5) Multiply.

= 6x 2 − 7x − 5 Combine like terms.

Use the FOIL Method to fi nd the product.

5. ( m + 5)( m − 6) 6. (x − 4)(x + 2)

7. (k + 5)(6k + 3) 8. ( 2u + 1

— 2

) ( u − 3

— 2

)

EXAMPLE Multiplying Binomials Using the FOIL Method33

Exercises 22–30

344 Chapter 7 Polynomial Equations and Factoring344 Chapter 7 Polynomial Equations and Factoring

In hockey, a goalie behind the goal line can only play a puck in a trapezoidal region.

a. Write a polynomial that represents the area of the trapezoidal region.

1

— 2

h(b1 + b2) = 1

— 2

(x − 7)[x + (x + 10)] Substitute.

= 1

— 2

(x − 7)(2x + 10) Combine like terms.

= 1

— 2

[2x 2 + 10x + (− 14x) + (− 70)] Use the FOIL Method.

= 1

— 2

(2x 2 − 4x − 70) Combine like terms.

= x 2 − 2x − 35 Distributive Property

b. Find the area of the trapezoidal region when the shorter base is 18 feet.

Find the value of x 2 − 2x − 35 when x = 18.

x 2 − 2x − 35 = 182 − 2(18) − 35 Substitute 18 for x.

= 324 − 36 − 35 Simplify.

= 253 Subtract.

The area of the trapezoidal region is 253 square feet.

Find the product.

9. ( x + 1)( x 2 + 5x + 8) 10. (n − 3)(n2 − 2n + 4)

11. WHAT IF? How does the polynomial in Example 5 change if the longer base is extended by 1 foot? Explain.

EXAMPLE Real-Life Application55

F O I L

x ft

(x + 10) ft

(x − 7) ft

Exercises 40–45

Find (x + 5)(x 2 − 3x − 2).

x 2 − 3x − 2 × x + 5 Align like terms vertically.

5x 2 − 15x − 10 Distributive Property

x 3 − 3x 2 − 2x Distributive Property

x 3 + 2x 2 − 17x − 10 Combine like terms.

The product is x 3 + 2x 2 − 17x − 10.

EXAMPLE Multiplying a Binomial and a Trinomial44

Multiply 5(x 2 − 3x − 2).

Multiply x(x 2 − 3x − 2).

Section 7.3 Multiplying Polynomials 345

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

1. VOCABULARY Describe two ways to fi nd the product of two binomials.

2. WRITING Explain how the letters of the word FOIL can help you remember how to multiply two binomials.

Write the product of the two binomials shown by the algebra tiles.

3. (x − 2)(x + 2) =

4. (− x + 3)(2x − 1) =

Use the Distributive Property to fi nd the product.

5. (x + 1)(x + 3) 6. ( y + 6)( y + 4) 7. (z − 5)(z + 3)

8. (a + 8)(a − 3) 9. (g − 7)(g − 2) 10. (n − 6)(n − 4)

11. (3m + 1)(m + 9) 12. (2p − 4)(3p + 2) 13. (6 − 5s)(2 − s)

14. ERROR ANALYSIS Describe and correct the (t − 2)(t + 5) = t − 2(t + 5) = t − 2t − 10 = −t − 10

✗error in fi nding the product.

15. CALCULATOR The width of a calculator can be represented by (3x + 1) inches. The length of the calculator is twice the width. Write a polynomial that represents the area of the calculator.

11

Exercises7.3Help with Homework

346 Chapter 7 Polynomial Equations and Factoring

Use a table to fi nd the product.

16. (x + 3)(x + 1) 17. ( y + 10)( y − 5) 18. (h − 8)(h − 9)

19. (− 3 + 2 j )(4 j − 7) 20. (5c + 6)(6c + 5) 21. (5d − 12)(− 7 + 3d)

Use the FOIL Method to fi nd the product.

22. (b + 3)(b + 7) 23. (w + 9)(w + 6) 24. (k + 5)(k − 1)

25. (x − 4)(x + 8) 26. (q − 3)(q − 4) 27. (z − 5)(z − 9)

28. (t + 2)(2t + 1) 29. (5v − 3)(2v + 4) 30. (9 − r)(2 − 3r)

31. ERROR ANALYSIS Describe and correct the error in fi nding the product.

(r + 6)(r − 7 ) = r (r) + r (7) + 6(r ) + 6(7) = r 2 + 7r + 6r + 42 = r 2 + 13r + 42

✗

32. OPEN-ENDED Write two binomials whose product includes the term 12.

33. SOCCER The soccer fi eld is rectangular.

a. Write a polynomial that represents the area of the soccer fi eld.

b. Use the polynomial in part (a) to fi nd the area of the fi eld when x = 90.

c. A groundskeeper mows 200 square yards in 3 minutes. How long does it take the grounds-keeper to mow the fi eld?

Write a polynomial that represents the area of the shaded region.

34.

y − 2

y − 3

35.

p + 1

2p − 6

36.

x + 2

x − 1x − 5

x − 4

Find the product.

37. (n + 3)(2n 2 + 1) 38. (x + y )(2x − y ) 39. (2r + s)(r − 3s)

40. (x − 4)(x 2 − 3x + 2) 41. (f 2 + 4f − 8)(f − 1) 42. (3 + i )(i 2 + 8i − 2)

43. (t 2 − 5t + 1)(− 3 + t) 44. (b − 4)(5b 2 − 5b + 4) 45. (3e 2 − 5e + 7)(6e + 1)

46. REASONING Can you use the FOIL method to multiply a binomial by a trinomial? a trinomial by a trinomial? Explain your reasoning.

22

33

(x + 10) yd

(x − 30) yd

44

Section 7.3 Multiplying Polynomials 347

47. AMUSEMENT PARK You go to an amusement park (x + 1) times each year and pay (x + 40) dollars each time, where x is the number of years after 2011.

a. Write a polynomial that represents your yearly admission cost.

b. What is your yearly admission cost in 2013?

48. PRECISION You use the Distributive Property to multiply (x + 3)(x − 5). Your friend uses the FOIL Method to multiply (x − 5)(x + 3). Should your answers be equivalent? Justify your answer.

49. REASONING The product of (x + m)(x + n) is x 2 + bx + c.

a. What do you know about m and n when c < 0?

b. What do you know about m and n when c > 0?

50. PICTURE You design the wooden picture frame and paint the front surface.

a. Write a polynomial that represents the area of wood you paint.

b. You design the picture frame to display a 5-inch by 8-inch photograph. How much wood do you paint?

51. The shipping container is a rectangular prism. Write a polynomial that represents the volume of the container.

Write the polynomial in standard form. Identify the degree and classify the polynomial by the number of terms. (Section 7.1)

52. 2x − 5x 2 − x 3 53. z 2 − 5

— 7

z 54. − 15y 7

55. MULTIPLE CHOICE Which system of linear equations does the graph represent? (Section 4.1)

○A y = 3x + 4 ○B y = 2x + 1

y = − 2x − 6 y = − x − 2

○C y = − x + 7 ○D y = x + 10

y = 4x − 8 y = − 3x + 2

x in.

(x 3) in.

(x 7) in.

(x 4) in.

(x + 2) ft

(x + 1) ft(4x − 3) ft

x

y

1234 2

1

2

3

4