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x 4 + x 3 + x 2 + x + 1 Let p(x) = x 4 + x 3 + x 2 + x + 1 The zero of x + 1 is - 1. p(- 1) = (- 1) 4 + (- 1) 3 + (- 1) 2 + (- 1) + 1 = 1 - 1 + 1 - 1 + 1 = 1 ≠ 0 ∴ By factor theorem, x + 1 is not a factor of x 4 + x 3 + x 2 + x + 1.

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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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102. The polynomials ax³+ 3x² - 3 and 2x³ - 5x + a when divided by x - 4 leave the same remainder in each case. 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:

x⁴ + x³ + x² + x + 1

x⁴ + x³ + x² + x + 1
Let p(x) = x⁴ + x³ + x² + x + 1
The zero of x + 1 is - 1.
p(- 1) = (- 1)⁴ + (- 1)³ + (- 1)² + (- 1) + 1
= 1 - 1 + 1 - 1 + 1 = 1 ≠ 0
∴ By factor theorem, x + 1 is not a factor of x⁴ + x³ + x² + x + 1.

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

x⁴ + 3x³ + 3x² + x + 1

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

left parenthesis straight x cubed minus straight x squared minus left parenthesis 2 plus square root of 2 right parenthesis straight x plus square root of 2 right parenthesis

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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

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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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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

104. Determine which of the following polynomials has (x + 1) a factor:x4 + x3 + x2 + x + 1

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

x⁴ + x³ + x² + x + 1