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Factor Theorem

On dividing, ...

Question

On dividing, $3 x^{3} + x^{2} + 2 x + 5$ by a polynomial g(x), the quotient and remainder are $3 x - 5$ and $9 x + 10$ respectively. Find g(x)

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Solution

By using division rule, we have
Divided = Quotient × Divisor + Remainder
$∴ 3 x^{3} + x^{2} + 2 x + 5 = (3 x - 5) g (x) + 9 x + 10 \Rightarrow 3 x^{3} + x^{2} + 2 x + 5 - 9 x - 10 = (3 x - 5) g (x) \Rightarrow 3 x^{3} + x^{2} - 7 x - 5 = (3 x - 5) g (x) \Rightarrow g (x) = \frac{3 x^{3} + x^{2} - 7 x - 5}{3 x - 5}$

$∴ g (x) = x^{2} + 2 x + 1$

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