Find $p$ and $q$ such that $(x + 1)$ and $(x - 1)$ are factors of the polynomial $x^{4} + p x^{3} - 3 x^{2} + q x + q$ .

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Solution

Let
$A = x^{4} + {p x}^{3} - {3 x}^{2} + q x + x$ with $(x + 1)$ & $(x - 1)$ are the 2 factors

$∴ x = 1 & x = - 1 s a t i s f i e s A$

Derivation of p
$\Rightarrow (- 1)^{4} - p - 3 - q + q = 0$
$\Rightarrow p = - 2$

Derivation of q
$\Rightarrow 1 + p - 3 + q + q = 0$
$\Rightarrow p + 2 q = 2$
$\Rightarrow 2 q = 4$
$\Rightarrow q = 2$

$H e n c e, (p, q) = (- 2, 2)$

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