On dividing polynomial $x^{3} - 3 x^{2} + x + 2$ 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

As we know that,
$Dividend = Divisor \times Quotient + Remainder . . . . . (1)$
According to the question,
$Dividend = f (x) = x^{3} - 3 x^{2} + x + 2$
$Divisor = g (x) = ?$
$Quotient = q (x) = x - 2$
$Remainder = r (x) = - 2 x + 4$
Now, from ${e q}^{n} (1)$ , we have
$f (x) = g (x) \times q (x) + r (x)$
$x^{3} - 3 x^{2} + x + 2 = g (x) \times (x - 2) + (- 2 x + 4)$
$(x^{3} - 3 x^{2} + x + 2) - (- 2 x + 4) = g (x) \times (x - 2)$
$\Rightarrow g (x) = \frac{(x^{3} - 3 x^{2} + 3 x - 2)}{(x - 2)}$
As from the attached image, we have
$(x^{3} - 3 x^{2} + 3 x - 2) \div (x - 2) = x^{2} - x + 1$
$∴ g (x) = \frac{(x^{3} - 3 x^{2} + 3 x - 2)}{(x - 2)} = (x^{2} - x + 1)$

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On dividing polynomial x3−3x2+x+2 by a polynomial g(x), the quotient and remainder were x−2 and −2x+4, respectively. Find g(x).

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