Question

Show that x-1 is a factor of x^10-1 and also of x^11-1

Answer 1

Let p(x)=x

10

−1 and q(x)=x

11

−1

If (x−1) is factor of x

10

−1 and x

11

−1 then the value of p(x)=x

10

−1 and q(x)=x

11

−1 is zero when x−1=0 or x=1

Replace x by 1 we get

p(x)=x

10

−1

p(1)=(1)

10

−1

p(1)=1−1=0

And

q(x)=x

11

−1

q(1)=(1)

11

−1

q(1)=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