Question

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

• A. $x^{2}+2x+1$
• B. $x^{2}-2x+1$
• C. $x+x+1$
• D. $x^{2}-x+1$

Hint:

With known quotient and remainder, recover the divisor via \(g(x)=\dfrac{f(x)-r(x)}{q(x)}\).

Answer 1

The Correct Option is D

Dividend $f(x)=x^{3}-3x^{2}+x+2$, quotient $q(x)=x-2$, remainder $r(x)=-2x+4$. Use $f(x)=g(x)q(x)+r(x)$: \[ g(x)=\frac{f(x)-r(x)}{q(x)} =\frac{x^{3}-3x^{2}+x+2-(-2x+4)}{x-2} =\frac{x^{3}-3x^{2}+3x-2}{x-2}. \] Divide: $(x-2)(x^{2}-x+1)=x^{3}-3x^{2}+3x-2$. Thus $g(x)=\boxed{x^{2}-x+1}$.