Question

Use the factor theorem to prove that x+c is a factor of x^n + c^n if n >_ 1 (greater than or equal to sign: >_ ) is an odd integer.

Answer 1

We need to prove that x+c is a factor of
\[x^n+c^n\ if\ n\ge 1\]
If x+c is a factor of \(f(x)=(x^n+c^n)\) then f(-c)=0
let's put x=-c
\[(-c)^n+c^n\]
We know that if n=even, \((-c)^n=c^n\)
and if n is odd \(=>(-c)^n=-c^n\)
so if n is odd
\[f(-c)=-c^n+c^n=0\]
Hence x+c is a factor of \(x^n+c^n\) for n=odd

Answer 2

@ap2332 did you understand?

Answer 3

yes, thank you. this is very helpful