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
ANSWERED
asked 2021-05-08
Use the factorization theorem to prove that x−c is a factor of \(x^{n}−c^{n}\) for any positive integer n.

Answers (1)

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 \(x^{n}-c^{n}\)
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