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6.3 Multiply Polynomials

Distributing a monomial across every term, the FOIL and vertical layouts for two binomials, and trinomial times binomial products.

Elementary Algebra 2e, 6.3 Multiply Polynomials

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105 exercises

6.3 Exercise 1

Before you get started, take this readiness quiz.

Distribute: 2(x+3).2 \left(x + 3\right) .
If you missed this problem, review the reference.

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6.3 Exercise 2

Combine like terms: x2+9x+7x+63.x^{2} + 9 x + 7 x + 63 .
If you missed this problem, review the reference.

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6.3 Exercise 4

Multiply: 5(x+7).5 \left(x + 7\right) .

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6.3 Exercise 5

Multiply: 3(y+13).3 \left(y + 13\right) .

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6.3 Exercise 7

Multiply: x(x−7).x \left(x - 7\right) .

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6.3 Exercise 8

Multiply: d(d−11).d \left(d - 11\right) .

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6.3 Exercise 10

Multiply: 5x(x+4y).5 x \left(x + 4 y\right) .

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6.3 Exercise 11

Multiply: 2p(6p+r).2 p \left(6 p + r\right) .

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6.3 Exercise 13

Multiply: −3y(5y2+8y−7).−3 y \left(5 y^{2} + 8 y - 7\right) .

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6.3 Exercise 14

Multiply: 4x2(2x2−3x+5).4 x^{2} \left(2 x^{2} - 3 x + 5\right) .

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6.3 Exercise 16

Multiply: 4x(3x2−5x+3).4 x \left(3 x^{2} - 5 x + 3\right) .

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6.3 Exercise 17

Multiply: −6a3(3a2−2a+6).−6 a^{3} \left(3 a^{2} - 2 a + 6\right) .

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6.3 Exercise 19

Multiply: (x+8)p.\left(x + 8\right) p .

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6.3 Exercise 20

Multiply: (a+4)p.\left(a + 4\right) p .

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6.3 Exercise 22

Multiply: (x+8)(x+9).\left(x + 8\right) \left(x + 9\right) .

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6.3 Exercise 23

Multiply: (5x+9)(4x+3).\left(5 x + 9\right) \left(4 x + 3\right) .

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6.3 Exercise 25

Multiply: (3b+5)(4b+6).\left(3 b + 5\right) \left(4 b + 6\right) .

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6.3 Exercise 26

Multiply: (a+10)(a+7).\left(a + 10\right) \left(a + 7\right) .

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6.3 Exercise 28

Multiply: (5y+2)(6y−3).\left(5 y + 2\right) \left(6 y - 3\right) .

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6.3 Exercise 29

Multiply: (3c+4)(5c−2).\left(3 c + 4\right) \left(5 c - 2\right) .

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6.3 Exercise 31

Multiply: (a+7)(a−b).\left(a + 7\right) \left(a - b\right) .

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6.3 Exercise 32

Multiply: (x+5)(x−y).\left(x + 5\right) \left(x - y\right) .

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6.3 Exercise 34

Multiply using the FOIL method: (x+6)(x+8).\left(x + 6\right) \left(x + 8\right) .

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6.3 Exercise 35

Multiply using the FOIL method: (y+17)(y+3).\left(y + 17\right) \left(y + 3\right) .

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6.3 Exercise 37

Multiply: (x−7)(x+5).\left(x - 7\right) \left(x + 5\right) .

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6.3 Exercise 38

Multiply: (b−3)(b+6).\left(b - 3\right) \left(b + 6\right) .

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6.3 Exercise 40

Multiply: (3x+7)(5x−2).\left(3 x + 7\right) \left(5 x - 2\right) .

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6.3 Exercise 41

Multiply: (4y+5)(4y−10).\left(4 y + 5\right) \left(4 y - 10\right) .

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6.3 Exercise 43

Multiply: (10c−d)(c−6).\left(10 c - d\right) \left(c - 6\right) .

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6.3 Exercise 44

Multiply: (7x−y)(2x−5).\left(7 x - y\right) \left(2 x - 5\right) .

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6.3 Exercise 46

Multiply: (x2+6)(x−8).\left(x^{2} + 6\right) \left(x - 8\right) .

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6.3 Exercise 47

Multiply: (y2+7)(y−9).\left(y^{2} + 7\right) \left(y - 9\right) .

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6.3 Exercise 49

Multiply: (2ab+5)(4ab−4).\left(2 a b + 5\right) \left(4 a b - 4\right) .

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6.3 Exercise 50

Multiply: (2xy+3)(4xy−5).\left(2 x y + 3\right) \left(4 x y - 5\right) .

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6.3 Exercise 52

Multiply using the Vertical Method: (5m−7)(3m−6).\left(5 m - 7\right) \left(3 m - 6\right) .

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6.3 Exercise 53

Multiply using the Vertical Method: (6b−5)(7b−3).\left(6 b - 5\right) \left(7 b - 3\right) .

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6.3 Exercise 55

Multiply using the Distributive Property: (y−3)(y2−5y+2).\left(y - 3\right) \left(y^{2} - 5 y + 2\right) .

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6.3 Exercise 56

Multiply using the Distributive Property: (x+4)(2x2−3x+5).\left(x + 4\right) \left(2 x^{2} - 3 x + 5\right) .

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6.3 Exercise 58

Multiply using the Vertical Method: (y−3)(y2−5y+2).\left(y - 3\right) \left(y^{2} - 5 y + 2\right) .

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6.3 Exercise 59

Multiply using the Vertical Method: (x+4)(2x2−3x+5).\left(x + 4\right) \left(2 x^{2} - 3 x + 5\right) .

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6.3 Exercise 60

Multiply.

4(w+10)4 \left(w + 10\right)

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6.3 Exercise 62

Multiply.

−3(a+7)−3 \left(a + 7\right)

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6.3 Exercise 64

Multiply.

2(x−7)2 \left(x - 7\right)

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6.3 Exercise 66

Multiply.

−3(k−4)−3 \left(k - 4\right)

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6.3 Exercise 68

Multiply.

q(q+5)q \left(q + 5\right)

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6.3 Exercise 70

Multiply.

−b(b+9)− b \left(b + 9\right)

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6.3 Exercise 72

Multiply.

−x(x−10)− x \left(x - 10\right)

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6.3 Exercise 74

Multiply.

6r(4r+s)6 r \left(4 r + s\right)

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6.3 Exercise 76

Multiply.

12x(x−10)12 x \left(x - 10\right)

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6.3 Exercise 78

Multiply.

−9a(3a+5)−9 a \left(3 a + 5\right)

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6.3 Exercise 80

Multiply.

3(p2+10p+25)3 \left(p^{2} + 10 p + 25\right)

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6.3 Exercise 82

Multiply.

−8x(x2+2x−15)−8 x \left(x^{2} + 2 x - 15\right)

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6.3 Exercise 84

Multiply.

5q3(q3−2q+6)5 q^{3} \left(q^{3} - 2 q + 6\right)

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6.3 Exercise 86

Multiply.

−8y(y2+2y−15)−8 y \left(y^{2} + 2 y - 15\right)

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6.3 Exercise 88

Multiply.

5q3(q2−2q+6)5 q^{3} \left(q^{2} - 2 q + 6\right)

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6.3 Exercise 90

Multiply.

−4z2(3z2+12z−1)−4 z^{2} \left(3 z^{2} + 12 z - 1\right)

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6.3 Exercise 92

Multiply.

(2m−9)m\left(2 m - 9\right) m

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6.3 Exercise 94

Multiply.

(w−6)·8\left(w - 6\right) \cdot 8

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6.3 Exercise 96

Multiply.

4(x+10)4 \left(x + 10\right)

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6.3 Exercise 98

Multiply.

15(r−24)15 \left(r - 24\right)

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6.3 Exercise 100

Multiply.

−3(m+11)−3 \left(m + 11\right)

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6.3 Exercise 102

Multiply.

−8(z−5)−8 \left(z - 5\right)

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6.3 Exercise 104

Multiply.

u(u+5)u \left(u + 5\right)

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6.3 Exercise 106

Multiply.

n(n2−3n)n \left(n^{2} - 3 n\right)

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6.3 Exercise 108

Multiply.

6x(4x+y)6 x \left(4 x + y\right)

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6.3 Exercise 110

Multiply.

5p(11p−5q)5 p \left(11 p - 5 q\right)

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6.3 Exercise 112

Multiply.

3(v2+10v+25)3 \left(v^{2} + 10 v + 25\right)

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6.3 Exercise 114

Multiply.

2n(4n2−4n+1)2 n \left(4 n^{2} - 4 n + 1\right)

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6.3 Exercise 116

Multiply.

−8y(y2+2y−15)−8 y \left(y^{2} + 2 y - 15\right)

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6.3 Exercise 118

Multiply.

5q3(q2−2q+6)5 q^{3} \left(q^{2} - 2 q + 6\right)

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6.3 Exercise 120

Multiply.

−4z2(3z2+12z−1)−4 z^{2} \left(3 z^{2} + 12 z - 1\right)

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6.3 Exercise 122

Multiply.

(2y−9)y\left(2 y - 9\right) y

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6.3 Exercise 124

Multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

(w+5)(w+7)\left(w + 5\right) \left(w + 7\right)

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6.3 Exercise 126

Multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

(p+11)(p−4)\left(p + 11\right) \left(p - 4\right)

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6.3 Exercise 128

Multiply the binomials. Use any method.

(x+8)(x+3)\left(x + 8\right) \left(x + 3\right)

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6.3 Exercise 130

Multiply the binomials. Use any method.

(y−6)(y−2)\left(y - 6\right) \left(y - 2\right)

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6.3 Exercise 132

Multiply the binomials. Use any method.

(w−4)(w+7)\left(w - 4\right) \left(w + 7\right)

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6.3 Exercise 134

Multiply the binomials. Use any method.

(p+12)(p−5)\left(p + 12\right) \left(p - 5\right)

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6.3 Exercise 136

Multiply the binomials. Use any method.

(6p+5)(p+1)\left(6 p + 5\right) \left(p + 1\right)

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6.3 Exercise 138

Multiply the binomials. Use any method.

(2t−9)(10t+1)\left(2 t - 9\right) \left(10 t + 1\right)

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6.3 Exercise 140

Multiply the binomials. Use any method.

(5x−y)(3x−6)\left(5 x - y\right) \left(3 x - 6\right)

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6.3 Exercise 142

Multiply the binomials. Use any method.

(a+b)(2a+3b)\left(a + b\right) \left(2 a + 3 b\right)

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6.3 Exercise 144

Multiply the binomials. Use any method.

(4z−y)(z−6)\left(4 z - y\right) \left(z - 6\right)

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6.3 Exercise 146

Multiply the binomials. Use any method.

(x2+3)(x+2)\left(x^{2} + 3\right) \left(x + 2\right)

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6.3 Exercise 148

Multiply the binomials. Use any method.

(x2+8)(x2−5)\left(x^{2} + 8\right) \left(x^{2} - 5\right)

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6.3 Exercise 150

Multiply the binomials. Use any method.

(5ab−1)(2ab+3)\left(5 a b - 1\right) \left(2 a b + 3\right)

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6.3 Exercise 152

Multiply the binomials. Use any method.

(6pq−3)(4pq−5)\left(6 p q - 3\right) \left(4 p q - 5\right)

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6.3 Exercise 154

Multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

(x+5)(x2+4x+3)\left(x + 5\right) \left(x^{2} + 4 x + 3\right)

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6.3 Exercise 156

Multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

(y+8)(4y2+y−7)\left(y + 8\right) \left(4 y^{2} + y - 7\right)

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6.3 Exercise 158

Multiply. Use either method.

(w−7)(w2−9w+10)\left(w - 7\right) \left(w^{2} - 9 w + 10\right)

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6.3 Exercise 160

Multiply. Use either method.

(3q+1)(q2−4q−5)\left(3 q + 1\right) \left(q^{2} - 4 q - 5\right)

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6.3 Exercise 162

Mixed Practice

(10y−6)+(4y−7)\left(10 y - 6\right) + \left(4 y - 7\right)

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6.3 Exercise 164

Mixed Practice

(x2−4x−34)−(x2+7x−6)\left(x^{2} - 4 x - 34\right) - \left(x^{2} + 7 x - 6\right)

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6.3 Exercise 166

Mixed Practice

5q(3q2−6q+11)5 q \left(3 q^{2} - 6 q + 11\right)

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6.3 Exercise 168

Mixed Practice

(s−7)(s+9)\left(s - 7\right) \left(s + 9\right)

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6.3 Exercise 170

Mixed Practice

(y2−2y)(y+1)\left(y^{2} - 2 y\right) \left(y + 1\right)

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6.3 Exercise 172

Mixed Practice

(3n−4)(n2+n−7)\left(3 n - 4\right) \left(n^{2} + n - 7\right)

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6.3 Exercise 174

Mixed Practice

(7p+10)(7p−10)\left(7 p + 10\right) \left(7 p - 10\right)

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6.3 Exercise 176

Mixed Practice

(4m2−3m−7)m2\left(4 m^{2} - 3 m - 7\right) m^{2}

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6.3 Exercise 178

Mixed Practice

(5a+7b)(5a+7b)\left(5 a + 7 b\right) \left(5 a + 7 b\right)

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6.3 Exercise 180

Mixed Practice

(4y+12z)(4y−12z)\left(4 y + 12 z\right) \left(4 y - 12 z\right)

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6.3 Exercise 182

Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17. Think of 18 as 20−220 - 2 and 17 as 20−320 - 3.

  1. ⓐ Multiply (20−2)(20−3)\left(20 - 2\right) \left(20 - 3\right) by the FOIL method.
  2. ⓑ Multiply 18·1718 \cdot 17 without using a calculator.
  3. ⓒ Which way is easier for you? Why?
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6.3 Exercise 184

Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?

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6.3 Exercise 186

Multiply the following:

(m−3)(m+3)(n−10)(n+10)(p−8)(p+8)\begin{matrix}\left(m - 3\right) \left(m + 3\right) \\ \left(n - 10\right) \left(n + 10\right) \\ \left(p - 8\right) \left(p + 8\right)\end{matrix}

Explain the pattern that you see in your answers.

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6.3 Exercise 188

Multiply the following:

(x−4)(x−4)(y−1)(y−1)(z−7)(z−7)\begin{matrix}\left(x - 4\right) \left(x - 4\right) \\ \left(y - 1\right) \left(y - 1\right) \\ \left(z - 7\right) \left(z - 7\right)\end{matrix}

Explain the pattern that you see in your answers.

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Adapted from Elementary Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Elementary Algebra 2e. Imported at revision 77a933c.