Question

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

Answer 1

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(x)` will be divisible by `(x-x_1)` 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(x) = 15x^12+41x^9+13x^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.

`color(white)()`
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(x)` must be expressible in the form `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:

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

None of these works, so `f(x)` has no rational zeros.

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`