Question

Find $a$ if $(x-a)$ is factor of $x^3-a{x}^2+2x+a-1$.

Answer 1

$\Rightarrow$ $(x-a)$ is factor of $x^3-a{x}^2+2x+a-1$

$\Rightarrow$ Then, $a$ is the zero of polynomial $p\left(x\right)=x^3-a{x}^2+2x+a-1$

$\Rightarrow$ $p\left(x\right)=x^3-a{x}^2+2x+a-1$

$\Rightarrow$ $p\left(a\right)=(a{)}^3-a(a{)}^2+2\left(a\right)+a-1$

$\Rightarrow$ $0=a^3-a^3+2a+a-1$

$\Rightarrow$ $a^3-a^3+2a+a-1=0$

$\Rightarrow$ $3a-1=0$

$\Rightarrow$ $3a=1$

$\therefore$ $a=\frac{1}{3}$