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

Jun 5, 2016

The FTOA tells us it has $12$ zeros but does not help us find them...

#### Explanation:

The fundamental theorem of algebra tells you that a polynomial of degree $n > 0$ in one variable has a Complex (possibly Real) zero.

A corollary of this, often stated as part of the FTOA is that a polynomial of degree $n > 0$ in one variable will have exactly $n$ Complex (possibly Real) zeros counting multiplicity.

To see why this follows, note that if we have one zero ${x}_{1}$, then $f \left(x\right)$ will be divisible by $\left(x - {x}_{1}\right)$ resulting in a polynomial of degree $n - 1$. If $n - 1 > 0$ then this will have a zero ${x}_{2}$, etc.

In our example:

$f \left(x\right) = 15 {x}^{12} + 41 {x}^{9} + 13 {x}^{2} - 10$

is of degree $12$, so has exactly $12$ zeros counting multiplicity.

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

$\textcolor{w h i t e}{}$
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 $1$ or $3$ negative Real zeros.

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

Any zero of $f \left(x\right)$ must be expressible in the form $\frac{p}{q}$ for integers $p , q$ with $p$ a divisor of the constant term $- 10$ and $q$ a divisor of the coefficient $15$ of the leading term.

That means that the only possible rational zeros are:

$\pm \frac{1}{15}$, $\pm \frac{2}{15}$, $\pm \frac{1}{5}$, $\pm \frac{1}{3}$, $\pm \frac{2}{5}$, $\pm \frac{2}{3}$ $\pm 1$, $\pm \frac{5}{3}$, $\pm 2$, $\pm 5$, $\pm 10$

None of these works, so $f \left(x\right)$ has no rational zeros.

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

$x \approx - 1.38923$

$x \approx 0.735661$

$x \approx - 0.775742 \pm 0.203247 i$

$x \approx - 0.487897 \pm 0.736584 i$

$x \approx 0.14955 \pm 0.928279 i$

$x \approx 0.714459 \pm 1.21625 i$

$x \approx 0.726412 \pm 0.459408 i$