If the polynomial $f (x) = x^{2} + x^{3} + 8 x^{2} + a x + b$ is exactly by $x^{2} + 1$ then find the values of $a$ and $b .$

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Solution

Remainder $= (a - 1) x + (b - 9) = 0$
$\Rightarrow a - 1 = 0, b - 9 = 0$
$∴ a = 1, b = 9$

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