Find the zetoes of cubic polynomials
$x^{3} - 5 x^{2} + 6 x$
without drawing the graph of the polynomial.

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Solution

$x^{3} - 5 x^{2} + 6 x$
Let $f (x) = x^{3} - 5 x^{2} + 6 x$
$\Rightarrow x (x^{2} - 5 x + 6)$
$\Rightarrow x (x^{2} - 2 x - 3 x + 6)$
$\Rightarrow x [x (x - 2) - 3 (x - 2)]$
$\Rightarrow x (x - 2) (x - 3)$
$\Rightarrow x = 0$ or $x = 2$ or $x = 3$
$∴$ Zeroes of the polynomial $f (x)$ are three.i.e. $0, 2$ and $3$

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