If one of the factors of the polynomial $(3 x^{3} + 25 x^{2} + 36 x - 36) is (x + 3),$ then find the other factors using synthetic division.

(x + 6)

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(3 x - 2)

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(3 x + 2)

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(x - 6)

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Solution

The correct options are
A $(x + 6)$
B $(3 x - 2)$
$p (x) = (3 x^{3} + 25 x^{2} + 36 x - 36)$
One factor = $(x + 3)$

Using synthetic division,
$\begin{matrix} x^{3} & x^{2} & x^{1} & x^{0} 3 & 25 & 36 & - 36 - 3 & ↓ & - 9 & - 48 & 36 3 & 16 & - 12 & | 0 \end{matrix}$

$∴ p (x) = (x + 3) (3 x^{2} + 16 x - 12) Now, (3 x^{2} + 16 x - 12) = (3 x^{2} + 18 x - 2 x - 12) = 3 x (x + 6) - 2 (x + 6) = (3 x - 2) (x + 6) Hence, x^{3} - 2 x^{2} + 9 x + 18 = (x + 3) (3 x - 2) (x + 6)$

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