On dividing the polynomial $f (x) = x^{3} - 3 x^{2} + x + 2$ by a polynomial $g (x)$ , the quotient $q (x)$ and remainder $r (x)$ are x-2 and -2x+4 respectively. Find the polynomial $g (x)$ . [4 MARKS]

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Solution

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

By division algorithm, we have

$f (x) = g (x) \times q (x) + r (x)$

$\Rightarrow g (x) \times q (x) = f (x) - r (x)$

$\Rightarrow g (x) (x - 2) = x^{3} - 3 x^{2} + x + 2 - (- 2 x + 4)$

$\Rightarrow g (x) (x - 2) = x^{3} - 3 x^{2} + 3 x - 2$

Thus, $g (x)$ is a factor of $x^{3} - 3 x^{2} + 3 x - 2$ other than the factor $(x - 2)$ .

So, to get $g (x)$ , we divide $x^{3} - 3 x^{2} + 3 x - 2$ by $(x - 2)$

Hence, $g (x) = x^{2} - x + 1$

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On dividing $x^{3} - 3 x^{2} + x + 2$ by a polynomial g(x), the quotient and remainder were $x - 2$ and $- 2 x + 4$ , respectively. Find g(x).

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On dividing $x^{3} - 3 x^{2} + x + 2$ by a polynomial g(x), the quotient and remainder were $x - 2$ and $- 2 x + 4$ , respectively. Find g(x).