Find the $H C F$ of the pair of polynomials using equations:
$x^{3} - 3 x^{2} + 4 x - 12, x^{4} + x^{3} + 4 x^{2} + 4 x$

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Solution

Factors of $x^{3} - 3 x^{2} + 4 x - 12$
$= x^{2} (x - 3) + 4 (x - 3)$
$= (x - 3) (x + 4)$
Factors of $x^{4} + x^{3} + 4 x^{2} + 4 x$
$= x^{3} (x + 1) + 4 x (x + 1)$
$= (x^{3} + 4 x) (x + 1)$
$= x (x + 4) (x + 1)$
H.C.F of $x^{3} - 3 x^{2} + 4 x - 12$ and $x^{4} + x^{3} + 4 x^{2} + 4 x$
Common factors of $(x - 3) (x + 4)$ and $x (x + 4) (x + 1)$
is $x + 4$
$∴$ H.C.F $= x + 4$

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