If the quotient obtained on dividing $(x^{4} + 10 x^{3} + 35 x^{2} + 50 x + 29)$ by $(x + 4)$ is $(x^{3} - a x^{2} + b x + 6)$ then find $a, b$ and also the remainder

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Solution

The division of $(x^{4} + 10 x^{3} + 35 x^{2} + 50 x + 29)$ and $(x + 4)$ is shown above:

Therefore, from the division, we get that the quotient is $(x^{3} + 6 x^{2} + 11 x + 6)$ and the remainder is $5$ .

But it is also given that the quotient is $(x^{3} - a x^{2} + b x + 6)$ , thus on comparing the coefficients of both the quotients $(x^{3} + 6 x^{2} + 11 x + 6)$ and $(x^{3} - a x^{2} + b x + 6)$ , we get

$- a x^{2} = 6 x^{2}$
$\Rightarrow$ $- a = 6$
$\Rightarrow$ $a = - 6$

Also,

$b x = 11 x$
$\Rightarrow$ $b = 11$

Hence, $a = - 6$ , $b = 11$ and the remainder is $5$ .

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