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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Solution

Here in the given question,

Dividend $= x^{3} - 3 x^{2} + x + 2$

Quotient $= x - 2$

Remainder $= - 2 x + 4$

Divisor = g(x)

We know that,

Dividend = Quotient $\times$ Divisor + Remainder

$\Rightarrow x^{3} - 3 x^{2} + x + 2 = (x - 2) \times g (x) + (- 2 x + 4) \Rightarrow x^{3} - 3 x^{2} + x + 2 - (- 2 x + 4) = (x - 2) \times g (x)$
$\Rightarrow x^{3} - 3 x^{2} + 3 x - 2 = (x - 2) \times g (x)$
$\Rightarrow g (x) = \frac{(x^{3} - 3 x^{2} + 3 x - 2)}{(x - 2)}$
$∴ g (x) = (x^{2} - x + 1)$

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