# Thread: Find the roots of this polynomial

1. ## Find the roots of this polynomial

This is highly urgent.

3x^5 - 19x^4 + 9x^3 + 71x^2 + 84x + 20 = 0.

We are looking for real solutions..
I believe there is only one - though, I have no idea how to find it. Horner's Method does not seem to be of use, neither do I seem to be able to rearrange anything in a beneficial way.

2. Originally Posted by Logic
This is highly urgent.

3x^5 - 19x^4 + 9x^3 + 71x^2 + 84x + 20 = 0.

We are looking for real solutions..
I believe there is only one - though, I have no idea how to find it. Horner's Method does not seem to be of use, neither do I seem to be able to rearrange anything in a beneficial way.
yes, there's one root. it is close to -0.31742. if you have the patience, you can use Newton's method to find a better approximation. i doubt it will be rational, but in the off-chance that it is, you might try using the Remainder theorem and/or Factor theorem

3. I am unfamiliar with either of these theorems and I was hoping to do better than approximations?

4. Originally Posted by Logic
I am unfamiliar with either of these theorems and I was hoping to do better than approximations?
by the theorems i mentioned, if a rational root exists, it is of the form $\pm \frac {\text{a factor of }20}{\text{a factor of }3}$. so find all of those and plug them in to see if you get zero.

if you find no such root, then there are no rational solutions. this being a quintic polynomial, there is no formula to find the solutions, and, in fact, the solutions cannot be given in terms of radicals, so the best you can do in that case is an approximation.