Question

If the polynomial $x^2-2x+k$ is a factor of $x^4-6x^3+16x^2-26x+10-a$, then find the value of $k$ and $a.$

Answer 1

After dividing polynomial $x^4-6x^3+16x^2-26x+10-x-a$ by $x^2-2x+k$ we got remainder, $(-10+2k)x+(10-a-8k+k^2).$

Which will be equal to zero if,

$-10+2k=0\text{and}10-a-8k+k^2=0$

For $-10+2k=0$

$\Rightarrow 2k=10$

$\Rightarrow k=5$

For $10-a-8k+k^2=0$

$\Rightarrow 10-a-8\times 5+25=0$

$\Rightarrow -5-a=0$

$\Rightarrow a=-5$

So, $k=5,a=-5$