Question

use factor theorm show that x-1 is factor of polynomial
${x}^{20} - 1$
and
${x}^{21} - 1$

Answer 1

Hey mate thank for the question
g(x)=x-1
x=1
f(x)=x^20-1
f(1)=1^20-1
f(x)=1-1=0

Answer 2

Factor theoram stated that if x-a is a factor of f(x)
then f(a)=0

so now come to your question
$x - 1 \: is \: factor \\ f(1) = 0 \\ f(x) = {x}^{20} - 1 \\ f(1) = 1 - 1 = 0 \\ hence \: proved \\ \\ for \: other \: part \\ f(x) = {x}^{21} - 1 \\ f(1) = 1 - 1 = 0 \\ proved \: again$