Find the $H . C . F .$ of the expressions $x^{3} + (a + b) x^{2} + (a b + 1) x + b$ and $x^{3} + 2 a x^{2} + (a^{2} + 1) x + a$ .

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Solution

1.)
$x^{3} + (a + b) x^{2} + (a b + 1) x + b$
$x^{3} + a x^{2} + b x^{2} + a b x + x + b$
$x (x^{2} + a x + 1) + b (x^{2} + a x + 1)$
$(x + b) (x^{2} + a x + 1)$

2.)
$x^{3} + 2 a x^{2} + (a^{2} + 1) x + a$
$x^{3} + 2 a x^{2} + a^{2} x + x + a$
$x (x^{2} + a x + 1) + a (x^{2} + a x + 1)$
$(x + a) (x^{2} + a x + 1)$
$∴ H C F = x^{2} + a x + 1$

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