Question

If the polynomial $6x^4+8x^3-5x^2+ax+b$ is exactly divisible by the polynomial $2x^2-5$, then find the value of $a$ and $b$.

Answer 1

Given polynomial

$p\left(x\right)=6x^4+8x^3-5x^2+ax+b$

And $g\left(x\right)=2x^2-5$

So,

On divide $p\left(x\right)$ to $g\left(x\right)$ and we get,

Quotient $=3x^2+4x+5$

Reminder$=(20+a)x+(25+b)$

Given that,

The polynomial $6x^4+8x^3-5x^2+ax+b$ is exactly divisible by the polynomial $2x^2-5$.

Hence,

$(20+a)x+(25+b)=0$

$\Rightarrow 20+a=0$

$\Rightarrow a=-20$

And $25+b=0$

$b=-25$

Hence, this is the answer.