# How do you find the roots for #f(x) = 15x^12 + 41x^9 + 13x^2 –10# using the fundamental theorem of algebra?

##### 1 Answer

The FTOA tells us it has

#### Explanation:

The fundamental theorem of algebra tells you that a polynomial of degree

A corollary of this, often stated as part of the FTOA is that a polynomial of degree

To see why this follows, note that if we have one zero

In our example:

#f(x) = 15x^12+41x^9+13x^2-10#

is of degree

That's all the FTOA tells you. It does not help you find the zeros.

**What else can we find out about the zeros of this polynomial?**

Note that the signs of the coefficients follow the pattern:

#+ + + -#

With one change of sign, that means that there will be exactly one positive Real zero.

Reversing the sign on the term of odd degree we get the pattern:

#+ - + -#

With three changes of sign, that means that there will be either

We can try to find rational zeros using the rational root theorem:

Any zero of

That means that the only possible rational zeros are:

#+-1/15# ,#+-2/15# ,#+-1/5# ,#+-1/3# ,#+-2/5# ,#+-2/3# #+-1# ,#+-5/3# ,#+-2# ,#+-5# ,#+-10#

None of these works, so

Ultimately, we are stuck with using numerical methods (e.g. Durand-Kerner) to find approximations for the zeros:

#x~~-1.38923#

#x~~0.735661#

#x~~-0.775742+-0.203247i#

#x~~-0.487897+-0.736584i#

#x~~0.14955+-0.928279i#

#x~~0.714459+-1.21625i#

#x~~0.726412+-0.459408i#