Question

If the polynomial $f\left(x\right)=x^4-6x^3+16x^2-25x+10$ is divided by another polynomial $x^2-2x+k$, the remainder comes out to be $x+a$, find k and a.
[4 MARKS]

Answer 1

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

By division algorithm, we have

Dividend = Divisor $\times$ Quotient + Remainder

$\Rightarrow$ Dividend - Remainder = Divisor $\times$ Quotient

$\Rightarrow$ Dividend - Remainder is always divisible by the divisor

It is given that $f\left(x\right)=x^4-6x^3+16x^2-25x+10$ when divided by $x^2-2x+k$ leaves $x+a$ as remainder.

$\therefore f\left(x\right)-(x+a)=x^4-6x^3+16x^2-26x+10-a$

is exactly divisible by $x^2-2x+k$

Let us now divide $x^4-6x^3+16x^2-26x+10-a$ by $x^2-2x+k$



For $f\left(x\right)-(x+a)=x^4-6x^3+16x^2-26x+10-a$ to be exactly divisible by $x^2-2x+k$,

we must have

$(-10+2k)x+(10-a-8k+k^2)=0$ for all x

$\Rightarrow -10+2k=0,10-a-8k+k^2=0$

$\Rightarrow k=5,10-a-40+25=0$

$\Rightarrow k=5anda=-5$