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Polynomials

Short Answer Type

121. Factorise
2y³ + y² - 2y - 1

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122. Use Factor Theorem to verify that x + a is a factor of xⁿ + aⁿ for any odd positive integer n.

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123. Determine the value of ‘b’ for which the polynomial 5x³ - x² + 4x + b is divisible by 1 - 5x.

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124. Find the values of a and b so that (x + 1) and (x - 1) are factors of x⁴ + ax³ - 3x² + 2x + b.

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125. For what value of a the polynomial 2x³ + ax² + 11x + a + 3 is exactly divisible by 2x - 1?

Let p(x) = 2x³ + ax² + 11x + a + 3
If p(x) is exactly divisible by 2x - 1, then by factor theorem,

$space space space space space space straight P open parentheses 1 half close parentheses equals 0 space space space space 12 straight x minus 1 equals 0 space rightwards double arrow space straight x space equals space 1 half rightwards double arrow space space space 2 open parentheses 1 half close parentheses cubed plus straight a open parentheses 1 half close parentheses squared plus 11 open parentheses 1 half close parentheses plus straight a plus 3 equals 0 rightwards double arrow space space space 1 fourth plus straight a over 4 plus 11 over 2 plus straight a plus 3 equals 0 rightwards double arrow space space space space fraction numerator 5 straight a over denominator 4 end fraction plus 35 over 4 equals 0 rightwards double arrow space space space space space straight a space equals space minus 7$