Question

Prove that a-b always divide a^n-b^n if n is natural number.​

Answer 1

Answer:

Let f(x,y)=xn−yn

Also, g(x,y)=x−y

By remainder theorem, we know that,

f(x,y)=g(x,y)⋅h(x,y)+k

where k is the remainder. Its degree is zero because g(x) is a linear equation, so the degree of the remainder must be one less than the degree of the divisor.

We need to prove that k=0 .

When x=y , we know that f(x,y)=g(x,y)=0

So, we see that 0=0⋅h(x,x)+k

⇒k=0

So, xn−yn is divisible by x−y .

I hope you know the remainder theorem.

Regards.