Question

If the polynomial $7x^3+ax+b$ is divisible by $x^2-x+1,$ then find the value of $2a+b$

Answer 1

Let $f\left(x\right)=7x^3+ax+b$ and $x^2-x+1=(x+\omega )(x+{\omega }^2)$

$\because f\left(x\right)$ is divisible by $x^2-x+1$

$\Rightarrow f(-\omega )=0$ and $f(-{\omega }^2)=0$

$\Rightarrow -7{\omega }^3-a\omega +b=d$ and $-7-a{\omega }^2+b=0$

$\Rightarrow -7-a\omega +b=d$ and $-7-a{\omega }^2+b=0$

On adding , we get

$-14-a(\omega +{\omega }^2)+2b=0$ ______ (1)

(OR) $-14+a+2b=0$ or $a+2b=14$

On subtracting , we get

$-a(\omega -{\omega }^2)=0$

$\Rightarrow a=0\because \omega -{\omega }^2\ne 0$

$\therefore 0+2b=14$

$\Rightarrow b=\frac{14}{2}=7$

$\therefore 2a+b=2\left(0\right)+7$

$2a+b=7$

Hence, $2a+b=7$