Factorize: $x^{3} - 4 x^{2} + 5 x - 2$

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Solution

Let $f (x) = x^{3} - 4 x^{2} + 5 x - 2$

Sum of the coefficients $= 1 - 4 + 5 - 2 = 0$

$∴ (x - 1)$ is a factor.

Remainder $= 0$

$f (x) = x^{3} - 4 x^{2} + 5 x - 2$

$= (x - 1) (x^{2} - 3 x + 2)$

$= (x - 1) (x - 1) (x - 2)$

$= {(x - 1)}^{2} (x - 2)$

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