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

Dec 19, 2015

You can't.

#### Explanation:

This theorem just tells you that a polynomial $P$ such that $\mathrm{de} g \left(P\right) = n$ 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 $\mathbb{C}$. Let's find its roots.

1st of all, you can factorize by $x$, so $f \left(x\right) = x \left({x}^{2} + 2 x - 24\right)$

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

You first calculate $\Delta = {b}^{2} - 4 a c = 4 + 4 \cdot 24 = 100 > 0$ 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, $f \left(x\right) = x \left(x + 6\right) \left(x - 4\right)$.

I hope it helped you.