Question

Use factor theorem to verify that (x-a) is always a factor of xⁿ-aⁿ​

Answer 1

Step-by-step explanation:

According to factor theorem, if x−k is a factor of a polynomial function f(x), then f(k)=0

Now to test for x+a being a factor of f(x)=xn+an,

as x+a=x−(−a), we must check for f(−a)

Now f(x)=xn+an

Hence, f(−a)=(−a)n+an

Observe if n is even then f(−a)=(−a)n+an=an+an=2an≠0 and hence x+a is not a factor of xn+an if n is even

but if n is odd then f(−a)=(−a)n+an=−an+an=0

and hence x+a is a factor of x