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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.

582 Views

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?

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126. Without actual division prove that x⁴ + 2x³ - 2x² + 2x - 3 is exactly divisible by x² + 2x - 3.

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127. Determine whether (3x — 2) is a factor of 3x³ + x² - 20x + 12?

Let p(x) = 3x³ + x² - 20x + 12
We see that

$straight P open parentheses 2 over 3 close parentheses equals 3 open parentheses 2 over 3 close parentheses cubed plus open parentheses 2 over 3 close parentheses squared minus 20 open parentheses 2 over 3 close parentheses plus 12 space space space space space space space space left enclose space space space 3 straight x minus 2 end enclose space equals space 0 space space space space space rightwards double arrow space straight x space 2 over 3 space space space space space space equals 8 over 9 plus 4 over 9 minus 40 over 3 plus 12 space space space space space space equals space 0$

Hence by converse of factor theorem, (3x - 2) is a factor of p(x).