Question

If $(x-1)$ and $(x-2)$ are factors of polynomial $p\left(x\right)=x^3+a{x}^2+bx+c$ and $a+b=1$, then the value of $b$ is

Answer 1

Since, $(x-1)$ is a factor of the given polynomial $p\left(x\right)$
Hence, $p\left(1\right)=0$
$\Rightarrow p\left(1\right)=a+b+c=-\mathrm{1.....}\left(i\right)$
but given that
$a+b=\mathrm{1.....}\left(ii\right)$
solving $\left(i\right)$ and $\left(ii\right)$, we get
$c=-2$
And,
$(x-2)$ is also a factor of the given polynomial $p\left(x\right)$
Hence, $p\left(2\right)=0$
$\Rightarrow p\left(2\right)=8+4a+2b-2=0$
$\Rightarrow 4a+2b+6=0$
$\Rightarrow 2a+2(a+b)+6=0$
$\Rightarrow 2a+8=0$
$\Rightarrow a=-4$
$\Rightarrow b=5$ (by solving $\left(ii\right))$