9 781292 040837

ISBN 978-1-29204-083-7

Beginning and Intermediate AlgebraLial Hornsby McGinnis

Fifth Edition

Beginning and Interm

ediate Algebra Lial et al. 5e

Pearson Education LimitedEdinburgh GateHarlowEssex CM20 2JEEngland and Associated Companies throughout the world

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ISBN 10: 1-269-37450-8ISBN 13: 978-1-269-37450-7

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library

Printed in the United States of America

Copyright_Pg_7_24.indd 1 7/29/13 11:28 AM

ISBN 10: 1-292-04083-1ISBN 13: 978-1-292-04083-7

ISBN 10: 1-292-04083-1ISBN 13: 978-1-292-04083-7

Multiplying Two Polynomials

Multiply

Combine like terms. NOW TRY= 4m5- 2m4

+ 24m3- 10m2

+ 20m

= 4m5- 2m4

+ 4m3+ 20m3

- 10m2+ 20m

= m214m32 + m21-2m22 + m214m2 + 514m32 + 51-2m22 + 514m2

1m2+ 5214m3

- 2m2+ 4m2

1m2+ 5214m3

- 2m2+ 4m2.

EXAMPLE 2NOW TRY

EXERCISE 2

Multiply.

1x2- 4212x2

- 5x + 32

NOW TRY ANSWERS

2. 2x4- 5x3

- 5x2+ 20x - 12

NOW TRY

EXERCISE 3

Multiply.

2t - 65t2

- 7t + 4

NOW TRY

EXERCISE 4

Find the product of

and .13 x2

-239x3

- 12x2+ 3

3. 10t3- 44t2

+ 50t - 244. 3x5

- 4x4- 6x3

+ 9x2- 2

Product NOW TRY3x4+ 11x3

+ 22x2+ 23x + 5

3x4+ 6x3

+ 12x2+ 3x 3x1x3

+ 2x2+ 4x + 12

5x3+ 10x2

+ 20x + 53x + 5

x3+ 2x2

+ 4x + 1

Multiplying Polynomials

To multiply two polynomials, multiply each term of the second polynomialby each term of the first polynomial and add the products.

Place like terms incolumns so theycan be added.

Write the polynomialsvertically

51x3+ 2x2

+ 4x + 12

This process is similar to multiplication of whole numbers.

Multiplying Polynomials Vertically

Multiply vertically.

Begin by multiplying each of the terms in the top row by 5.

Now multiply each term in the top row by 3x. Then add like terms.

5x3+ 10x2

+ 20x + 53x + 5

x3+ 2x2

+ 4x + 1

3x + 5x3

+ 2x2+ 4x + 1

1x3+ 2x2

+ 4x + 1213x + 52

EXAMPLE 3

Multiply each term of the secondpolynomial by each term of the first.

Multiplying Polynomials with Fractional Coefficients Vertically

Find the product of and

Terms of top row are multiplied by .

Terms of top row are multiplied by .

Add. NOW TRY

We can use a rectangle to model polynomial multiplication. For example, to find

label a rectangle with each term as shown next on the left. Then put the product ofeach pair of monomials in the appropriate box, as shown on the right.

2 2

1 1 23x2x 2x 6x2 4x

3x 3x

12x + 1213x + 22,

2m5- m4

+ 12m3- 5m2

+ 10m

2m5- m4

+ 2m3 12 m2

10m3- 5m2

+ 10m 52

12 m2

+52

4m3- 2m2

+ 4m

4m3- 2m2

+ 4m 12 m2

+52 .

EXAMPLE 4

Exponents and Polynomials

274

The product of the binomials is the sum of the four monomial products.

This approach can be extended to polynomials with any number of terms.

OBJECTIVE 3 Multiply binomials by the FOIL method. When multiplyingbinomials, the FOIL method reduces the rectangle method to a systematic approachwithout the rectangle. Consider this example.

Distributive property

Distributive property again

Multiply.

Combine like terms.

The letters of the word FOIL originate as shown.

The outer product, and the inner product, should be added mentally to get so that the three terms of the answer can be written without extra steps.

= x2+ 8x + 15

1x + 321x + 52

8x3x,5x,

1x + 321x + 52

1x + 321x + 52

1x + 321x + 52

1x + 321x + 52

= x2+ 8x + 15

= x2+ 3x + 5x + 15

= x1x2 + 31x2 + x152 + 3152

= 1x + 32x + 1x + 325

1x + 321x + 52

= 6x2+ 7x + 2

= 6x2+ 4x + 3x + 2

12x + 1213x + 22

Multiplying Binomials by the FOIL Method

Step 1 Multiply the two First terms of the binomials to get the first term ofthe answer.

Step 2 Find the Outer product and the Inner product and add them (whenpossible) to get the middle term of the answer.

Step 3 Multiply the two Last terms of the binomials to get the last term ofthe answer.

Add.8x5x3x

1x + 321x + 52

L = 15F = x2

IO

Exponents and Polynomials

Multiply the First terms: F

Multiply the Outer terms: OThis is the outer product.

Multiply the Inner terms: IThis is the inner product.

Multiply the Last terms: L3152.

31x2.

x152.

x1x2.

275

Using the FOIL Method

Use the FOIL method to find the product

Step 1 F Multiply the First terms:

Step 2 O Find the Outer product: .

I Find the Inner product: .

Add the outer and inner products mentally:

Step 3 L Multiply the Last terms:

First Last

Shortcut:

Inner

Outer

NOW TRY

1x + 821x - 62

1x + 821x - 62 = x2+ 2x - 48

81-62 = -48.

-6x + 8x = 2x.

81x2 = 8x

x1-62 = -6x

x1x2 = x2.

1x + 821x - 62.

EXAMPLE 5NOW TRY

EXERCISE 5

Use the FOIL method to findthe product.

1t - 621t + 52

NOW TRY ANSWERS

5. t2- t - 30

Using the FOIL Method

Find each product.

(a)F O I L

Multiply.

Combine like terms.

(b)

FOIL= 21p2- pq - 2q2

17p + 2q213p - q2

= 2k2+ 11ky + 15y2

= 2k2+ 6ky + 5ky + 15y2

= 2k1k2 + 2k13y2 + 5y1k2 + 5y13y2

12k + 5y21k + 3y2

EXAMPLE 7

NOW TRY

EXERCISE 6

Multiply.

17y - 3212x + 52

NOW TRY

EXERCISE 7

Find each product.

(a)

(b) 5x213x + 121x - 52

13p - 5q214p - q2

6. 14yx + 35y - 6x - 157. (a)

(b) 15x4- 70x3

- 25x2

12p2- 23pq + 5q2

Using the FOIL Method

Multiply

FirstOuterInnerLast

F O I LThe product NOW TRY19x - 2213y + 12 is 27xy + 9x - 6y - 2.

-219x - 2213y + 12-6y19x - 2213y + 12

9x19x - 2213y + 12 27xy19x - 2213y + 12

19x - 2213y + 12.

EXAMPLE 6

These unliketerms cannotbe combined.

NOW TRY

NOTE Alternatively, Example 7(c) can be solved as follows.

= 6x4- 10x3

- 24x2

= 12x3- 6x2213x + 42

2x21x - 3213x + 42

(c)

FOIL

= 6x4- 10x3

- 24x2

= 2x213x2- 5x - 122

2x21x - 3213x + 42

Distributiveproperty

Add the terms foundin Steps 1–3.

Exponents and Polynomials

Add. 2x

-6x

8x

1x + 821x - 62-48x2

Same answer

Multiply and first.

Multiply that product and .3x + 4

x - 32x2

276

5 EXERCISES

Concept Check In Exercises 1 and 2, match each product in Column I with the correct poly-nomial in Column II.

I II

1. (a)

(b)

(c)

(d) 1-6x32315x723-5x716x32

5x316x72 A.

B.

C.

D. -30x10

-216x9

30x10

125x21

I II

2. (a)

(b)

(c)

(d) 1x + 521x - 42

1x - 521x - 42

1x + 521x + 42

1x - 521x + 42 A.

B.

C.

D. x2+ x - 20

x2- x - 20

x2- 9x + 20

x2+ 9x + 20

Find each product. See Objective 1.

3. 4. 5.

6. 7. 8.

9. 10. 11.

12. 13. 14.

Find each product. See Example 1.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Find each product. See Examples 2–4.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

Find each product. Use the FOIL method. See Examples 5–7.

41. 42. 43.

44. 45. 46.

47. 48. 49.

50. 51. 52.

53. 54. 55.

56. 57. 58.

59. 60. 61.

62. 63. 64. 15a + 3b215a - 4b213x + 2y215x - 3y215x + 7213y - 82

14x + 3212y - 1212m - 3n21m + 5n213t - 4s21t + 3s2

16 - 5x212 + x215 - 3x214 + x218 - 3a212 + a2

16 - 5m212 + 3m21b + 8216b - 2215a + 1212a + 72

14m + 3214m + 3213x - 2213x - 22110 + r2110 - r2

19 + t219 - t213y + 5218y - 6212x + 3216x - 42

1 y + 821 y - 821x + 521x - 521t - 321t + 82

1n - 121n + 421n + 921n + 321m + 721m + 52

18y6+ 4y4

- 12y22a34

y2+ 2b16x4

- 4x2+ 8x2a

12

x + 3b

12m2+ m - 321m2

- 4m + 5215x2+ 2x + 121x2

- 3x + 52

12a + 321a4- a3

+ a2- a + 1212x - 1213x5

- 2x3+ x2

- 2x + 32

12y + 8213y4- 2y2

+ 1214m + 3215m3- 4m2

+ m - 52

12r - 1213r2+ 4r - 4219y - 2218y2

- 6y + 12

19a + 2219a2+ a + 1216x + 1212x2

+ 4x + 12

2p2q13p2q2- 5p + 2q227m3n213m2

+ 2mn - n32

4z318z2+ 5zy - 3y223a212a2

- 4ab + 5b22

-9a51-3a6- 2a4

+ 8a22-4r31-7r2+ 8r - 92

2m416 + 5m + 3m222y313 + 2y + 5y42

-7y13 + 5y2- 2y32-8z12z + 3z2

+ 3z32

4x13 + 2x + 5x323p1-2p3+ 4p22

4x15x + 322m13m + 22

17t5213t421- t8214x3212x221-x52x2 # 3x3 # 2x

y5 # 9y # y49r31-2s22-6m313n22

4a313b225p13q22-3m61-5m42

-15a41-2a5210p215p325y413y72

Exponents and Polynomials

277

65. 66.

67. 68.

Find polynomials that represent (a) the area and (b) the perimeter of each square or rectangle.

69. 70.

Find each product. In Exercises 81–84, 89, and 90, apply the meaning of exponents.

71. 72.

73. 74. 75.

76. 77. 78.

79. 80. 81.

82. 83. 84.

85. 86.

87. 88.

89. 90.

91. 92.

93. 94.

The figures in Exercises 95–98 are composed of triangles, squares, rectangles, and circles.Find a polynomial that represents the area of each shaded region. In Exercises 97 and 98,leave in your answers.

95. 96.

97. 98.

p

-2x513x2+ 2x - 5214x + 22 -4x313x4

+ 2x2- x21-2x + 12

3p312p2+ 5p21 p3

+ 2p + 12 5k21k3- 321k2

- k + 42

13r - 2s24 12z - 5y24714m - 3212m + 12 513k - 7215k + 22

-3a13a + 121a - 42 -4r13r + 2212r - 52

1 p - 323 12a + 123 13m + 12315k + 3q22 18m + 3n22 1m - 5231b - 1021b + 102 12p - 522 13m - 1221x + 722 1m + 622 1a - 421a + 42

a3p +

54

qb a2p -

53

qb a2x +

23

yb a3x -

34

yb

-8r315r2+ 2215r2

- 22 -5t412t4+ 1212t4

- 12

3y312y + 321 y - 52 2x212x - 521x + 32

3y + 7

y + 1 6x + 2

x + 7

x + 7x

xx

2x + 5

x + 14

3

3

xx

5x + 1

2x + 3

Apply a power rule for exponents. See Section 1.

99. 100. 101.

102. 103. 104.1-5a22 14x222 18y32213m22 15p22 1-2r22

PREVIEW EXERCISES

Exponents and Polynomials

278