Question

# Use the factorization theorem to prove that x−c is a factor of x^{n}−c^{n} for any positive integer n.

Polynomial factorization
Use the factorization theorem to prove that x−c is a factor of $$x^{n}−c^{n}$$ for any positive integer n.

2021-05-09
Step 1
Let's understand the factor theorem before we get into solving the question directly.
(x - a) is a factor of f(x) if and only if f(a) = 0
Step 2
In our case, $$f(x) = x^{n}-c^{n}$$
Basis factor theorem, (x - c) will be a factor of $$f(x) = x^{n}-c^{n}$$ if and only if f(c) = 0
Let's check:
$$f(c) = c^{n} - c^{n} = 0$$
Step 3
Hence, f(c) = 0
Hence, (x - c) is a factor of $$x^{n}-c^{n}$$