CBSE Class 11-science Answered

to prove that x-1 is a factor of x^2n-1 - 1 , it is sufficient to prove that x^2n-1 - 1 is divisible by x-1

we will prove it by mathematical induction

for n=1 , x-1 is divisible by x-1

for n=2 , x³-1 = (x-1)(x²+x+1) which is also divisible by x-1

let us assume that  x^2n-1 - 1 is divisible by x-1 for all n N

now we have to show that it is also true for n= N+1

for n=N+1

x^2N+1 - 1 = x² x^2N-1 - 1 + x² -x² = x² (x^2N-1 - 1) + (x² - 1)

x^2N-1 - 1 is divisible by x-1 ( true by our hypothesis)

and x² - 1 = (x-1)(x+1) which is also divisible by x-1

so it is true for n= N+1

i.e. x^2N-1 - 1 is divisible by x-1 or it has a factor equal to x-1