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p(x) = 2x 3 + x 2 - 2x - 1, g(x) = x + 1 g(x) = 0 ⇒ x + 1 = 0 ⇒ x = - 1 ∴ Zero of g(x) is - 1. Now, p(- 1) = 2(- 1) 3 + (- 1) 2 - 2(- 1) - 1 = - 2 + 1 + 2 - 1 = 0 ∴ By factor theorem, g(x) is a factor of p(x).

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Polynomials

Short Answer Type

101. If the polynomial p(x) = x⁴- 2x³ + 3x² - ax + 8 is divided by (x - 2), it leaves a remainder 10. Find the value of a.

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103. Determine which of the following polynomials has (x + 1) a factor:

x³ + x² + x + 1

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104. Determine which of the following polynomials has (x + 1) a factor:

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105. Determine which of the following polynomials has (x + 1) a factor:

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106. Determine which of the following polynomials has (x + 1) a factor:

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107. Use the factor theorem to determine whether g(x) is a factor ofp(x) in each of the following cases:

p(x) = 2x³ + x² - 2x - 1, g(x) = x + 1

p(x) = 2x³ + x² - 2x - 1, g(x) = x + 1
g(x) = 0
⇒ x + 1 = 0 ⇒ x = - 1
∴ Zero of g(x) is - 1.
Now, p(- 1)
= 2(- 1)³ + (- 1)² - 2(- 1) - 1
= - 2 + 1 + 2 - 1 = 0
∴ By factor theorem, g(x) is a factor of p(x).

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108. Use the factor theorem to determine whether g(x) is a factor ofp(x) in each of the following cases:

p(x) = x³ + 3x² + 3x + 1,g(x) = x + 2

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109. Use the factor theorem to determine whether g(x) is a factor ofp(x) in each of the following cases:

p(x) = x³ - 4x² + x + 6, g(x) = x - 3

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110. Find the value of k, if x - 1 is a factor of p(x) in each of the following cases:
p(x) = x² + x + k

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Test Yourself

Short Answer Type

107. Use the factor theorem to determine whether g(x) is a factor ofp(x) in each of the following cases: p(x) = 2x3 + x2 - 2x - 1, g(x) = x + 1

107. Use the factor theorem to determine whether g(x) is a factor ofp(x) in each of the following cases:

p(x) = 2x³ + x² - 2x - 1, g(x) = x + 1