Show that $(x + 4), (x - 3)$ and $(x - 7)$ are factors of $x^{3} - 6 x^{2} - 19 x + 84$

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Solution

$x^{3} - 6 x^{2} - 19 x + 84$
$= x^{3} + 4 x^{2} - 10 x^{2} - 40 x + 21 x + 84$
$= x^{2} (x + 4) - 10 (x + 4) + 21 (x + 4)$
$= (x + 4) (x^{2} - 10 + 21)$
$= (x + 4) (x^{2} - 7 x - 3 x + 21)$
$= (x + 4) [x (x - 7) - 3 (x - 7)]$
$= (x + 4)) (x - 3) (x - 7)$

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Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x).

f(x) = $x^{3} - 6 x^{2}$ - 19x + 84, g(x) = x - 7

(x - 1)

4 x^{3} - 6 x^{2} + 9 x - 7

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