Question

Question 4
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).

Answer 1

Here in the given question,

Dividend $=x^3-3x^2+x+2$

Quotient $=x-2$

Remainder $=-2x+4$

Divisor = g(x)

We know that,

Dividend = Quotient $\times$ Divisor + Remainder

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

$\therefore g\left(x\right)=(x^2-x+1)$