Question

What must be added to $f\left(x\right)=4x^4+2x^3-2x^2+x-1$ so that the resulting polynomial is divisible by $g\left(x\right)=x^2+2x-3$
[4 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

By division algorithm, we have

$f\left(x\right)=g\left(x\right)\times q\left(x\right)+r\left(x\right)$

$\Rightarrow f\left(x\right)-r\left(x\right)=g\left(x\right)\times q\left(x\right)$

$\Rightarrow f\left(x\right)+(-r(x\left)\right)$ = g(x) \times q(x)\)

Clearly, RHS is divisible by $g\left(x\right)$. Therefore, LHS is also divisible by $g\left(x\right)$. Thus, if we add $-r\left(x\right)$ to $f\left(x\right)$, then the resulting polynomial is divisible by $g\left(x\right)$. Let us now find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$.



$\therefore r\left(x\right)=-61x+65$

Hence, we should add $(-r(x)=61x-65$ to $f\left(x\right))$ so that the resulting polynomial is divisible by $g\left(x\right)$.