6.4 Multiplication of Polynomials and Special · PDF fileMultiplication of Polynomials and Special Products 6.4 6.4 OBJECTIVES 1. Evaluate f(x) g(x) for a given x 2. Multiply two polynomial

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403

Multiplication of Polynomials and Special Products

6.4

6.4 OBJECTIVES

1. Evaluate f(x) � g(x) for a given x2. Multiply two polynomial functions3. Square a binomial4. Find the product of two binomials as a difference

of squares

In Section 1.4, you saw the first exponent property and used that property to multiplymonomials. Let’s review.

Example 1

Multiplying Monomials

Multiply.

(8x2y)(4x3y4) � (8 � 4)(x2�3)(y1�4)

� 32x5y5

NOTEaman � am�n

Notice the use of the associativeand commutative properties to“regroup” and “reorder” thefactors.

Multiply.

Add exponents.

C H E C K Y O U R S E L F 1

Multiply.

(a) (4a3b)(9a3b2) (b) (�5m3n)(7mn5)

We now want to extend the process to multiplying polynomial functions.

Example 2

Multiplying a Monomial and a Binomial Function

Given f(x) � 5x2 and g(x) � 3x2 � 5x, and letting h(x) � f(x) � g(x), find h(x).

h(x) � f(x) � g(x)

� 5x2 � (3x2 � 5x) Apply the distributive property.

� 5x2 � 3x2 � 5x2 � 5x

� 15x4 � 25x3


Given f(x) � 3x2 and g(x) � 4x2 � x, and letting h(x) � f(x) � g(x), find h(x).

We can check this result by comparing the values of h(x) and of f(x) � g(x) for a specificvalue of x. This is illustrated in Example 3.

404 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS

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Example 3

Multiplying a Monomial and a Binomial Function

Given f(x) � 5x2 and g(x) � 3x2�5x, and letting h(x) � f(x) � g(x), compare f(1) � g(1)with h(1).

f(1) � g(1) � 5(1)2 � (3(1)2 � 5(1))

� 5(3 � 5)

� 5(�2)

� �10

From Example 2, we know that

h(x) � 15x4 � 25x3

So

h(1) � 15(1)4 � 25(1)3

� 15 � 25

� �10

Therefore, h(1) � f(1) � g(1).


Given f(x) � 3x2 and g(x) � 4x2 � x, and letting h(x) � f(x) � g(x), compare f (1) � g(1)with h(1).

The distributive property is also used to multiply two polynomial functions. To considerthe pattern, let’s start with the product of two binomial functions.

Example 4

Multiplying Binomial Functions

Given f(x) � x � 3 and g(x) � 2x � 5, and letting h(x) � f(x) � g(x), find the following.

(a) h(x)

h(x) � f(x) � g(x)

� (x � 3)(2x � 5) Apply the distributive property.

� (x � 3)(2x) � (x � 3)(5) Apply the distributive property again.

� (x)(2x) � (3)(2x) � (x)(5) � (3)(5)

� 2x2 � 6x � 5x � 15

� 2x2 � 11x � 15

Notice that this ensures that each term in the first polynomial is multiplied by each termin the second polynomial.

�

MULTIPLICATION OF POLYNOMIALS AND SPECIAL PRODUCTS SECTION 6.4 405©

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(b) f(1) � g(1)

f(1) � g(1) � (1 � 3)(2(1)� 5)

� 4(7)

� 28

(c) h(1)

From part (a), we have h(x) � 2x2 � 11x � 15, so

h(1) � 2(1)2 � 11(1) � 15

� 2 � 11 � 15

� 28

Again, we see that h(1) � f(1) � g(1).


Given f (x) � 3x � 2 and g(x) � x � 3, and letting h(x) � f(x) � g(x), find the following.

(a) h(x) (b) f(1) � g(1) (c) h(1)

Certain products occur frequently enough in algebra that it is worth learning specialformulas for dealing with them. Consider these products of two equal binomial factors.

(a � b)2 � (a � b)(a � b)

� a2 � 2ab � b2 (1)

(a � b)2 � (a � b)(a � b)

� a2 � 2ab � b2 (2)

We can summarize these statements as follows.

NOTEa2 � 2ab � b2

and

a2 � 2ab � b2

are called perfect-squaretrinomials.

Example 5

Squaring a Binomial

Find each of the following binomial squares.

(a) (x � 5)2 � x2 � 2(x)(5) � 52

� x2 � 10x � 25

NOTE Be sure to write out theexpansion in detail.

Square of first term Twice the productof the two terms

Square of last term

The square of a binomial has three terms, (1) the square of the first term, (2)twice the product of the two terms, and (3) the square of the last term.

(a � b)2 � a2 � 2ab � b2

and

(a � b)2 � a2 � 2ab � b2

Rules and Properties: Squaring a Binomial


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(b) (2a � 7)2 � (2a)2 � 2(2a)(7) � (�7)2

� 4a2 � 28a � 49

(a � b)(a � b) � a2 � b2

In words, the product of two binomials that differ only in the signs of theirsecond terms is the difference of the squares of the two terms of the binomials.

Rules and Properties: Product of Binomials Differing in Sign


Find each of the following products.

(a) (y � 5)(y � 5) (b) (2x � 3)(2x � 3) (c) (4r � 5s2)(4r � 5s2)

C A U T I O N

Be Careful! A very commonmistake in squaring binomials isto forget the middle term!

(y � 7)2

is not equal to

y2 � (7)2

The correct square is

y2 � 14y � 49

The square of a binomial isalways a trinomial.


Find each of the following binomial squares.

(a) (x � 8)2 (b) (3x � 5)2

Another special product involves binomials that differ only in sign. It will be extremelyimportant in your work later in this chapter on factoring. Consider the following:

(a � b)(a � b) � a2 � ab � ab � b2

� a2 � b2

Example 6

Finding a Special Product

Multiply.

(a) (x � 3)(x � 3) � x2 � (3)2

� x2 � 9

(b) (2x � 3y)(2x � 3y) � (2x)2 � (3y)2

� 4x2 � 9y2

(c) (5a � 4b2)(5a � 4b2) � (5a)2 � (4b2)2

� 25a2 � 16b4

When multiplying two polynomials that don’t fit one of the special product patterns,there are two different ways to set up the multiplication. Example 7 will illustrate the ver-tical approach.

C A U T I O N

The entire term 2x is squared,not just the x.

NOTE This format ensures thateach term of one polynomialmultiplies each term of theother.

MULTIPLICATION OF POLYNOMIALS AND SPECIAL PRODUCTS SECTION 6.4 407©

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Example 7

NOTE Again, this ensures thateach term of one polynomialmultiplies each term of theother.

Multiplying Polynomials

Multiply 3x3 � 2x2 � 5 and 3x � 2.

Step 1 3x3 � 2x2 � 53x � 2 Multiply by 2.

6x3 � 4x2 � 10

Step 2 3x3 � 2x2 � 53x � 2

6x3 � 4x2 � 109x4 � 6x3 � 15x

Step 3 3x3 � 2x2 � 53x � 2

6x3 � 4x2 � 109x4 � 6x3 � 15x Add the partial products.

9x4 � 4x2 � 15x � 10


Find the following product, using the vertical method.

(4x3 � 6x � 7)(3x � 2)

A horizontal approach to the multiplication in Example 7 is also possible by the distribu-tive property. As we see in Example 8, we first distribute 3x over the trinomial and then wedistribute 2 over the trinomial.

Example 8


Multiply (3x � 2)(3x3 � 2x2 � 5), using a horizontal format.

Step 1

(3x � 2)(3x3 � 2x2 � 5)

Step 2

� 9x4 � 6x3 � 15x � 6x3 � 4x2 � 10 Combine like terms.

Step 1 Step 2

� 9x4 � 4x2 � 15x � 10 Write the product in descendingform.

� �

Multiply by 3x. Note that we alignthe terms in the partial product.


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Find the product of Check Yourself 7, using a horizontal format.

Multiplication sometimes involves the product of more than two polynomials. In suchcases, the associative property of multiplication allows us to regroup the factors to makethe multiplication easier. Generally, we choose to start with the product of binomials. Exam-ple 9 illustrates this approach.

Example 9


Find the products.

(a) x(x � 3)(x � 3) � x(x2 � 9) Find the product (x � 3)(x � 3).

� x3 � 9x Then distribute x as the last step.

(b) 2x(x � 3)(2x � 1) � 2x(2x2 � 5x � 3) Find the product of the binomials.

� 4x3 � 10x2 � 6x Then distribute 2x.


Find each of the following products.

(a) m(2m � 3)(2m � 3) (b) 3a(2a � 5)(a � 3)

C H E C K Y O U R S E L F A N S W E R S

1. (a) 36a6b3; (b) �35m4n6 2. h(x) � 12x4 � 3x3 3. f(1) � g(1) � 15 � h(1)

4. (a) h(x) � 3x2 � 7x � 6; (b) f(1) � g(1) � 4; (c) h(1) � 4

5. (a) x2 � 16x � 64; (b) 9x2 � 30x � 25

6. (a) y2 � 25; (b) 4x2 � 9; (c) 16r2 � 25s4 7. 12x4 � 8x3 � 18x2 � 9x � 14

8. 12x4 � 8x3 � 18x2 � 9x � 14 9. (a) 4m3 � 9m; (b) 6a3 � 3a2 � 45a

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Exercises

In exercises 1 to 6, find each product.

1. (4x)(5y) 2. (�3m)(5n)

3. (6x2)(�3x3) 4. (5y4)(3y2)

5. (5r2s)(6r3s4) 6. (�8a2b5)(�3a3b2)

In exercises 7 to 14, f(x) and g(x) are given. Let h(x) � f(x) � g(x). Find (a) h(x), (b) f(1) � g(1), and (c) use the result of (a) to find h(1).

7. f(x) � 3x and g(x) � 2x2 � 3x 8. f(x) � 4x and g(x) � 2x2 � 7x

9. f(x) � �5x and g(x) � �3x2 � 5x � 8 10. f(x) � 2x2 and g(x) � �7x2 � 2x

11. f(x) � 4x3 and g(x) � 9x2 � 3x � 5 12. f(x) � 2x3 and g(x) � 2x3 � 4x

13. f(x) � 3x3 and g(x) � 5x2 � 4x 14. f(x) � �x2 and g(x) � �7x3 � 5x2

In exercises 15 to 24, find each product.

15. (x � y)(x � 3y) 16. (x � 3y)(x � 5y)

17. (x � 2y)(x � 7y) 18. (x � 7y)(x � 3y)

19. (5x � 7y)(5x � 9y) 20. (3x � 5y)(7x � 2y)

21. (7x � 5y)(7x � 4y) 22. (9x � 7y)(3x � 2y)

23. (5x2 � 2y)(3x � 2y2) 24. (6x2 � 5y2)(3x2 � 2y)

6.4

Name

Section Date

ANSWERS

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3. 4.

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In exercises 25 to 38, multiply polynomial expressions using the special product formulas.

25. (x � 5)2 26. (x � 7)2

27. (2x � 3)2 28. (5x � 3)2

29. (4x � 3y)2 30. (7x � 5y)2

31. (4x � 3y2)2 32. (3x3 � 7y)2

33. (x � 3y)(x � 3y) 34. (x � 5y)(x � 5y)

35. (2x � 3y)(2x � 3y) 36. (5x � 3y)(5x � 3y)

37. (4x2 � 3y)(4x2 � 3y) 38. (7x � 6y2)(7x � 6y2)

In exercises 39 to 42, multiply using the vertical format.

39. (3x � y)(x2 � 3xy � y2) 40. (5x � y)(x2 � 3xy � y2)

41. (x � 2y)(x2 � 2xy � 4y2) 42. (x � 3y)(x2 �3xy � 9y2)

In exercises 43 to 46, simplify each function.

43. f(x) � x(x � 3)(x � 1) 44. f(x) � x(x � 4)(x � 2)

45. f(x) � 2x(x � 5)(x � 4) 46. f(x) � x2(x � 4)(x2 � 5)

Multiply the following.

47. 48.

49. [x � (y � 2)][x � (y � 2)] 50. [x � (3 � y)][x � (3 � y)]

�x

3�

3

4��3x

4�

3

5��x

2�

2

3��2x

3�

2

5�

ANSWERS

25.

26.

27.

28.

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32.

33.

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35.

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If the polynomial p(x) represents the selling price of an object, then the polynomial R(x),in which R(x) � x � p(x), is the revenue produced by selling x objects. Use this informationto solve exercises 51 and 52.

51. If p(x) � 100 � 0.2x, find R(x). 52. If p(x) � 250 � 0.5x, find R(x).Find R(50). Find R(20).

In exercises 53 to 56, label the statements as true or false.

53. (x � y)2 � x2 � y2 54. (x � y)2 � x2 � y2

55. (x � y)2 � x2 � 2xy � y2 56. (x � y)2 � x2 � 2xy � y2

57. Area. The length of a rectangle is given by 3x � 5 centimeters (cm) and the width isgiven by 2x � 7 cm. Express the area of the rectangle in terms of x.

58. Area. The base of a triangle measures 3y � 7 inches (in.) and the height is 2y � 3 in.Express the area of the triangle in terms of y.

59. Revenue. The price of an item is given by p � 10 � 3x. If the revenue generated isfound by multiplying the number of items x sold by the price of an item, find thepolynomial that represents the revenue.

60. Revenue. The price of an item is given by p � 100 � 2x2. Find the polynomial thatrepresents the revenue generated from the sale of x items.

61. Tree planting. Suppose an orchard is planted with trees in straight rows. If there are5x � 4 rows with 5x � 4 trees in each row, how many trees are there in the orchard?

62. Area of a square. A square has sides of length 3x � 2 centimeters (cm). Express thearea of the square as a polynomial.

3y � 7

2y � 3

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52.

53.

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63. Area of a rectangle. The length and width of a rectangle are given by two consecutiveodd integers. Write an expression for the area of the rectangle.

64. Area of a rectangle. The length of a rectangle is 6 less than three times the width.Write an expression for the area of the rectangle.

65. Work with another student to complete this table and write the polynomial. A paperbox is to be made from a piece of cardboard 20 inches (in.) wide and 30 in. long. Thebox will be formed by cutting squares out of each of the four corners and folding upthe sides to make a box.

If x is the dimension of the side of the square cut out of the corner, when the sides arefolded up, the box will be x inches tall. You should use a piece of paper to try this tosee how the box will be made. Complete the following chart.

Write general formulas for the width, length, and height of the box and a generalformula for the volume of the box, and simplify it by multiplying. The variable willbe the height, the side of the square cut out of the corners. What is the highest powerof the variable in the polynomial you have written for the volume? Extend the table todecide what the dimensions are for a box with maximum volume. Draw a sketch ofthis box and write in the dimensions.

66. Complete the following statement: (a � b)2 is not equal to a2 � b2 because. . . . But,wait! Isn’t (a � b)2 sometimes equal to a2 � b2? What do you think?

67. Is (a � b)3 ever equal to a3 � b3? Explain.

Length of Side of Length of Width of Depth of Volume ofCorner Square Box Box Box Box

1 in.

2 in.

3 in.

n in.

30 in.

20 in.

x

ANSWERS

63.

64.

65.

66.

67.

412

68. In the following figures, identify the length and the width of the square, and then findthe area.

Length � ________

Width � ________

Area � ________

Length � ________

Width � ________

Area � ________

69. The square shown is x units on a side. The area is _______.

Draw a picture of what happens when the sides are doubled. The area is _______.

Continue the picture to show what happens when the sides are tripled. The area is_______.

If the sides are quadrupled, the area is _______.

In general, if the sides are multiplied by n, the area is _______.

If each side is increased by 3, the area is increased by _______.

If each side is decreased by 2, the area is decreased by _______.

In general, if each side is increased by n, the area is increased by _______, and ifeach side is decreased by n, the area is decreased by _______.

For each of the following problems, let x represent the number, then write an expressionfor the product.

70. The product of 6 more than a number and 6 less than that number.

71. The square of 5 more than a number.

72. The square of 4 less than a number.

73. The product of 5 less than a number and 5 more than that number.

x

x

x

x

2

2

a

a

b

b

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68.

69.

70.

71.

72.

73.

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Note that (28)(32) � (30 � 2)(30 � 2) � 900 � 4 � 896. Use this pattern to find each ofthe following products.

74. (49)(51) 75. (27)(33)

76. (34)(26) 77. (98)(102)

78. (55)(65) 79. (64)(56)

Answers1. 20xy 3. �18x5 5. 30r5s5 7. (a) 6x3 � 9x2; (b) �3; (c) �39. (a) 15x3 � 25x2 � 40x; (b) 0; (c) 0 11. (a) 36x5 � 12x4 � 20x3; (b) 28; (c) 2813. (a) 15x5 � 12x4; (b) 3; (c) 3 15. x2 � 4xy � 3y2

17. x2 � 5xy � 14y2 19. 25x2 � 80xy � 63y2 21. 49x2 � 63xy � 20y2

23. 15x3 � 10x2y2 � 6xy � 4y3 25. x2 � 10x � 25 27. 4x2 � 12x � 929. 16x2 � 24xy � 9y2 31. 16x2 � 24xy2 � 9y4 33. x2 � 9y2

35. 4x2 � 9y2 37. 16x4 � 9y2 39. 3x3 � 8x2y � 6xy2 � y3

41. x3 � 8y3 43. x3 � 2x2 � 3x 45. 2x3 � 2x2 � 40x

47. 49. x2 � y2 � 4y � 4 51. 100x � 0.2x2, 4500

53. False 55. True 57. 6x2 � 11x � 35 cm2 59. 10x � 3x2

61. 25x2 � 40x � 16 63. x(x � 2) or x2 � 2x 65.

67. 69. x2; 4x2; 9x2; 16x2; n2x2; 6x � 9; 4x � 4; 2xn � n2; 2xn � n2

71. x2 � 10x � 25 73. x2 � 25 75. 891 77. 9996 79. 3584

x2

3�

11x

45�

4

15

ANSWERS

74.

75.

76.

77.

78.

79.

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Documents

6.4 Multiplication of Polynomials and Special · PDF fileMultiplication of Polynomials and Special Products 6.4 6.4 OBJECTIVES 1. Evaluate f(x) g(x) for a given x 2. Multiply two polynomial