Questions on factor theorem.?

Question

1. What must be the value of m so that x+y is a factor of x^m - y^m?

2. Show that x-t is a factor of x^n - t^n for any positive integer n.

3. Show that x+t is a factor of x^n + t^n for any positive odd integer n.

I don't know how to do no.1 using this theorem, although I do know the answer. And are there any special things to note for nos. 2 and 3, or do I simply show that if P(x) is the polynomial and x-t is a factor, then P(t) = 0?

Thanks!

Answer 1

Your idea is correct. We are using the Factor Theorem which states that x - k is a factor of f(x) iff f(k) = 0.

1. Let f(x) = x^m - y^m, and consider f(-y). We have f(-y) =
(-y)^m - y^m. This is 0 iff m is even, and then and only then
x - (-y) = x + y is a factor of f.

2. As you noted, f(t) = t^n - t^n = 0 for all positive integers n.

3. As above, f(-t) = (-t)^n + t^n = 0 whenever n is odd.

Answer 2

1) Let P(x)=x^m -y^m. For x+y to be a factor, P(-y) must be 0, so you need (-y)^m -y^m=0=(-1)^m y^m -y^m=
[(-1)^m -1] y^m.

Hence, either y=0 or m is even [so (-1)^m=1].

2) +3) Yes, just show that P(t)=0 for an appropriate P(x). Remember that the factor theorem goes both ways: the polynomial is 0 at a root r if and only if (x-r) is a factor.

Answer 3

1 note a^2-b^2=(a+b)(a-b) so m=2n
2 wrong note a^3-b^3=(a+b)(a^2-ab- b^2) use induction
3 wrong see 2 and 1