Find if polynomial $x^{4} - 3 x^{3} + 7 x^{2} - 8 x + 12$ is exactly divisible by $x^{2} - 2 x + 2$ .

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Solution

$\frac{x^{4} - 3 x^{3} + 7 x^{2} - 8 x + 12}{x^{2} - 2 x + 2}$

$= x^{2} + \frac{- x^{3} + 5 x^{2} - 8 x + 12}{x^{2} - 2 x + 2}$

$= x^{2} - x + \frac{3 x^{2} - 6 x + 12}{x^{2} - 2 x + 2}$

$= x^{2} - x + 3 + \frac{6}{x^{2} - 2 x + 2}$

$∴ \frac{(x^{4} - 3 x^{3} + 7 x^{2} - 8 x + 12)}{(x^{2} - 2 x + 2)} = x^{2} - x + 3 + \frac{6}{x^{2} - 2 x + 2}$

It's not a perfect division.

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