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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.

Let p(x) = 5x³ - x² + 4x + b
1 - 5x = 0

$rightwards double arrow space space space space space space space space 5 straight x space equals space 1 space space space rightwards double arrow space space space straight x space equals space 1 fifth$

$therefore$ The zero of 1 - 5x is $1 fifth$
If p(x) is divisivle by 1 - 5x, then $straight p open parentheses 1 fifth close parentheses equals 0$
| By Factor Theorem
$rightwards double arrow space 5 open parentheses 1 fifth close parentheses cubed minus open parentheses 1 fifth close parentheses squared plus 4 open parentheses 1 fifth close parentheses plus straight b equals 0 rightwards double arrow space space 1 over 25 minus 1 over 25 plus 4 over 5 plus straight b equals 0 rightwards double arrow space space space 4 over 5 plus straight b equals 0 rightwards double arrow space space space space straight b space equals space minus 4 over 5.$