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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

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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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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

Let space straight p left parenthesis straight x right parenthesis equals straight x cubed minus straight x squared minus left parenthesis 2 plus square root of 2 right parenthesis left parenthesis negative 1 right parenthesis plus square root of 2 space space space space space equals space minus 1 minus 1 plus 2 plus square root of 2 plus square root of 2 equals 2 square root of 2 not equal to 0

∴ By factor theorem, x + 1 is not a factor of

straight x to the power of 4 minus straight x cubed space left parenthesis 2 plus square root of 2 right parenthesis straight x plus square root of 2

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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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