By which polynomial we should divide $(4 x^{4} - 5 x^{3} - 39 x^{2} - 46 x - 2)$ to get quotient as $(x^{2} - 3 x - 5)$ and remainder as $(- 5 x + 8)$ .

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Solution

$f (x) = (4 x^{4} - 5 x^{3} - 39 x^{2} - 4 x - 2)$ be the divident
$g (x)$ be the divider
$q (x) = x^{2} - 3 x - 5$ be the quotient and
$r (x) = - 5 x + 8$ be the remainder
We know that
$f (x) = g (x) q (x) + r (x)$
$\Rightarrow f (x) - r (x) = g (x) q (x)$
$\frac{f (x) - r (x)}{q (x)} = g (x)$
$\frac{4 x^{4} - 5 x^{3} - 39 x^{2} - 46 x - 2 - (- 5 x + 9)}{x^{2} - 3 x - 5} = g (x)$
$\Rightarrow 4 x^{4} - 5 x^{3} - 39 x 2 - 14 x - 10 \div x^{2} - 3 x - 5$
By doing division; we get
$x^{2} - 3 x - 5) 4 x^{4} - {5 x}^{3} - {39 x}^{2} - 41 x - 10 ({4 x}^{2} + 7 x + 2 {4 x}^{2} - {12 x}^{3} - {20 x}^{2}______________{7 x}^{3} - 19 x^{2} - 41 x - 10 {7 x}^{3} - {21 x}^{2} - 35 x_{2 x}^{2} - 6 x - 10 {2 x}^{2} - 6 x - 10____________0$
$∴ g (x) = 4 x^{2} + 7 x + 2$

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