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

Step 1: Since \(\deg f=3\) and \(\deg q=1\), \(g(x)\) is a monic quadratic; let \(g(x)=x^2+bx+c\).

Step 2: Expand \((x^2+bx+c)(x-2)+(-2x+4)\) and match with \(x^3-3x^2+x+2\): \[ b-2=-3\Rightarrow b=-1,\qquad 4-2c=2\Rightarrow c=1 \]

Step 3: \[ g(x)=\boxed{x^2-x+1} \]