is (x^2-5) a factor of x^4-3x^2-1?

Answer 1

Let's compute the remainder of the polynomial after division by x^2 - 5. So, we want to evaluate:

x^4-3x^2-1 Mod (x^2 - 5)

If we use the equals sign to mean equality Modulo x^2 - 5 in the following, then we have:

x^2 = 5 ---->

x^4 = 25

So, we have:

x^4 - 3 x^2 - 1 =

25 - 15 - 1 = 9

This is nonzero, so x^2 - 5 is not a factor of the polynomial.