When $x^{5} - 5 x^{4} + 9 x^{3} - 6 x^{2} - 16 x + 13$ is divided by $x^{2} - 3 x + a$ , the quotient and remainder are $x^{3} - 2 x^{2} + x + 1$ and $- 15 x + 11$ , respectively. Find the value of $a$ .

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Solution

Step 1: Given information
Dividend: $x^{5} - 5 x^{4} + 9 x^{3} - 6 x^{2} - 16 x + 13$
Divisor: $x^{2} - 3 x + a$
Quotient: $x^{3} - 2 x^{2} + x + 1$
Remainder: $- 15 x + 11$
Step 2: Determine the value of $a$ .
We know that, $Dividend = (Divisor \times Quotient) + Remainder$ .
$\begin{array}{rcl} (x^{5} - 5 x^{4} + 9 x^{3} - 6 x^{2} - 16 x + 13) & = & (x^{2} - 3 x + a) (x^{3} - 2 x^{2} + x + 1) + (- 15 x + 11) \\ x^{5} - 5 x^{4} + 9 x^{3} - 6 x^{2} - x + 2 & = & x^{5} - 5 x^{4} + 7 x^{3} - 2 x^{2} - 3 x + a x^{3} + a x - 2 a x^{2} + a \\ 2 x^{3} - 4 x^{2} + 2 x + 2 - a & = & a x^{3} - 2 a x^{2} - a x + a \end{array}$
Equating the coefficients of like terms, we get
$a = 2$
Hence, the value of $a$ is $2$ .

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