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

We know the formula,

Dividend = Divisor x quotient + Remainder

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

Substitute the values, we get

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

Add $2 x$ and subtract $4$ both side,

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

Simplify and divide by $x - 2$

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

$g (x) = \frac{(x - 2) (x^{2} - x + 1)}{x - 2}$

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

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