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 $a$ and $b$ . Also find the remainder.

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Solution

$x^{4} + 10 x^{3} + 35 x^{2} + 50 x + 29 = (x^{4} + 10 x^{3} + 25 x^{2} + 50 x + 24) + 5$
Given : $(x + 4)$ is one of the factor of $x^{4} + 10 x^{3} + 35 x^{2} + 50 x + 24$
And by using Synthetic Division we get another factor as $(x^{3} + 6 x^{3} + 11 x + 6)$ and the remainder $5$
$∴$ $a = - 6$ , $b = 11$ and the remainder is $5$ .

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