Question

How do you find the number of roots for `f(x) = x^3 + 2x^2 - 24x` using the fundamental theorem of algebra?

Answer 1

You can't.

Explanation:

This theorem just tells you that a polynomial `P` such that `deg(P) = n` has at most `n` different roots, but `P` can have multiple roots. So we can say that `f` has at most 3 different roots in `CC`. Let's find its roots.

1st of all, you can factorize by `x`, so `f(x) = x(x^2 + 2x - 24)`

Before using this theorem, we need to know if P(x) = `(x^2 + 2x - 24)` has real roots. If not, then we will use the fundamental theorem of algebra.

You first calculate `Delta = b^2 - 4ac = 4 + 4*24 = 100 > 0` so it has 2 real roots. So the fundamental theorem of algebra is not of any use here.

By using the quadratic formula, we find out that the two roots of P are `-6` and `4`. So finally, `f(x) = x(x+6)(x-4)`.

I hope it helped you.