Find the quotient and remainder when $x^{5} - 5 x^{4} + 9 x^{3} - 6 x^{2} - 16 x + 13$ is divided by $x^{2} - 3 x + 2$ .

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Solution

Let us solve the given equation by long division of polynomials method
$x^{3} - 2 x^{2} + x + 1$
$x^{2} - 3 x + 2) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{5} - 5 x^{4} + 9 x^{3} - 6 x^{2} - 16 x + 13$
$x^{5} - 3 x^{4} + 2 x^{3}$
$(-) (+) (-) - ----------------- -$
$- 2 x^{4} + 7 x^{3} - 6 x^{2}$
$- 2 x^{4} + 6 x^{3} - 4 x^{2}$
$(+) (-) (+) - ----------------- -$
$x^{3} - 2 x^{2} - 16 x$
$x^{3} - 3 x^{2} + 2 x$
$(-) (+) (-) - ----------------- -$
$x^{2} - 18 x + 13$
$x^{2} - 3 x + 2$
$(-) (+) (-) - ----------------- -$
$- 15 x + 11$
Quotient is $x^{3} - 2 x^{2} + x + 1$
Remainder is $- 15 x + 11$

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