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Synthetic Division of Polynomials

If the polyno...

Question

If the polynomial $(x^{4} + {2 x}^{3} + {8 x}^{2} + 12 x + 18)$ is divided by $(x^{2} + 5),$ the remainder comes out to be $(p x + q)$ ,find the value of $p$ and $q$ .

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Solution

$x^{2} + 5) x^{4} + 2 x^{3} + 8 x^{2} + 12 x + 18 (x^{2} + 2 x + 3$
$x^{4}$ $5 x^{2}$
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$2 x^{3} + 3 x^{2}$
$2 x^{3} + 10 x$

$3 x^{2} + 2 x$
$3 x^{2} + 15$
_
$2 x + 3$
$2 x + 3 = p * + q$
$p = 2$
$q = 3$ .

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