# How do you find the roots for #f(x) = 17x^15 + 41x^12 + 13x^3 - 10# using the fundamental theorem of algebra?

##### 1 Answer

The FTOA does not help you find the zeros - it only tells you that this polynomial of degree

#### Explanation:

By roots, I will assume you mean zeros, i.e. values of

The so called fundamental theorem of algebra (FTOA) is neither fundamental nor a theorem of algebra, but what it does tell you is that any non-zero polynomial in one variable with Complex coefficients has a zero in

A simple corollary of this - often stated as part of the FTOA - is that a polynomial in one variable of degree

In our example,

The FTOA does not help you actually find the zeros.

**Bonus**

What else can we find out about the zeros of this

Note that the coefficients of

Note

So the positive Real zero is in

#f(-x) = -17x^5+41x^12-13x^3-10#

has two changes of sign, so

We find:

#f(-1) = -17+41-13-10 = 1 > 0#

So there is a Real zero in

Any other zeros will occur in Complex conjugate pairs, since all of the coefficients of

Note that all of the degrees are multiples of

#g(t) = 17t^5+41t^4+13t-10#