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

1 Answer
Topscooter
Dec 19, 2015

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.

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