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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.

Let p(x) = xⁿ + aⁿThe zero of x + a is - a. | x + a = 0 ⇒ x = - a
Now,
p(- a) = (- a)ⁿ + aⁿ = (- 1)ⁿaⁿ + aⁿ
= (- 1)aⁿ + aⁿ

$because$ n is an odd positive interger
$therefore$ (-1)ⁿ = - 1
= - aⁿ + aⁿ= 0
∴ By Factor Theorem, x + a is a factor of xⁿ + aⁿ for any odd positive integer n.