Question

Show that $(x-1)$ is a factor of $x^{10}-1$ and also of $x^{11}-1$

Answer 1

Let $p\left(x\right)=x^{10}-1$ and $q\left(x\right)=x^{11}-1$

If $(x-1)$ is factor of $x^{10}-1$ and $x^{11}-1$ then the value of $p\left(x\right)=x^{10}-1$ and $q\left(x\right)=x^{11}-1$ is zero when $x-1=0$ or $x=1$

Replace $x$ by 1 we get

$p\left(x\right)=x^{10}-1$

$p\left(1\right)=(1{)}^{10}-1$

$p\left(1\right)=1-1=0$

And

$q\left(x\right)=x^{11}-1$

$q\left(1\right)=(1{)}^{11}-1$

$q\left(1\right)=1-1=0$

So p(1) and q(1) are zero

then $x-1$ is a factor of $x^{10}-1$ and $x^{11}-1$