Factorise the following polynomial using synthetic division:
$- 7 x + 3 + 4 x^{3}$ .

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Solution

$p (x) = - 7 x + 3 + 4 x^{3}$
$= 4 x^{3} - 7 x + 3$
$p (1) = 4 (1)^{3} - 7 (1) + 3$
$= 4 - 7 + 3$
$= 7 - 7$
$= 0$
$∴ x - 1$ is a factor
$4 x^{2} + 4 x - 3 = 4 x^{2} + 6 x - 2 x - 3$
$= 2 x (2 x + 3) - 1 (2 x + 3)$
$= (2 x + 3) (2 x - 1)$
$∴$ The factors of $- 7 x + 3 + 4 x^{3} = (x - 1) (2 x + 3) (2 x - 1)$

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