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

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

p(x) = $$x^3 – 3x^2 + x + 2$$

q(x) = x – 2 and r(x) = -2x + 4

Solution : By division algorithm, we know that

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

Therefore, $$x^3 – 3x^2 + x + 2$$ = (x – 2) $$\times$$ g(x) + (-2x + 4)

$$\implies$$ $$x^3 – 3x^2 + x + 2 + 2x – 4$$ = (x – 2) $$\times$$ g(x)

$$\implies$$  g(x) = $$x^3 – 3x^2 + 3x – 2\over x – 2$$

On dividing  $$x^3 – 3x^2 + x + 2$$ by x – 2, we get g(x)

Hence, g(x) = $$x^2 – x + 1$$.