Chapter 4.5, Problem 107AYU

To determine

The term $x-c$ is factor of $x^n-c^n$ for any positive integer $n$ is to be proved with the help of the factor theorem.

Explanation of Solution

Given:

  $f\left(x\right)=x^n-c^n$

Proof:

According to the factor theorem:

If $x-k$ is factor of $f\left(x\right)$ for any positive integer $n$ then the value of $f\left(k\right)$ equals to zero.

Now, need to take test $x-c$ being a factor of $f\left(x\right)=x^n-c^n$ . Check for $\begin{array}{l}x-c=0\\ x=c\end{array}$

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

Hence, proved.