Question

The number of distinct common root(s) of the equations $x^5-x^3+x^2-1=0$ and $x^4-1=0$ is

Answer 1

$x^5-x^3+x^2-1=0\cdots \left(1\right)x^4-1=0\cdots \left(2\right)$
Subracting $x\times \left(2\right)$ from $\left(1\right)$, we get
$-x^3+x^2+x-1=0\Rightarrow -x^2(x-1)+1(x-1)=0\Rightarrow (x-1)(1-x^2)=0\Rightarrow x=1\text{or}{x}^2=1\therefore x=\pm 1$

Hence, the number of distinct common roots is $2$.