Question

Use the Factor Theorem to prove that $x-c$ is a factor of

$x^n-c^n$ for any positive integer n.

Short answer

If $f\left(x\right)=x^n-c^n$then $f\left(c\right)=0$so $x-c$is a factor of$x^n-c^n$

Step-by-step solution

Step 1. given data

The given expression is

$x^n-c^n$

Factor is$x-c$

Step 2. proof

Consider function $f\left(x\right)=x^n-c^n$

According to Factor Theorem, If $x-c$is a factor of function then $f\left(c\right)=0$

substitute c for x in the function

$$\begin{gathered} f\left(x\right)=x^n-c^n \\ f\left(c\right)=c^n-c^n \\ f\left(c\right)=0 \end{gathered}$$

the function value is zero for $x=c$

so$x-c$is a factor of$x^n-c^n$