Representing a Quintic Polynomial as the Product of its Linear Factors

Question

I have the polynomial [ f(x) = 3x^5 + x^4 + 15x^3 + 12x + 4 ] and I'm trying to represent it as the product of its linear factors. I've tried the Rational Root Theorem, but it seems there are no rational roots. I also attempted to factor by grouping, but that didn't yield any results either.

I've been told that the roots of this polynomial might not be expressible in terms of radicals. Is there a way to determine the roots, even if they're in a more complex form? Additionally, I've heard of Tschirnhausen transformations and Galois theory in this context. Can anyone provide insights or methods to tackle this problem using these or any other approaches?

Answer 1

This all depends on exactly where your polynomial lives. Is it a polynomial over an algebraically closed field? Note also that, with results from Galois theory (and now, many other places), polynomials of degree $\geq 5$ generally can't be solved by radicals.

If you're really dead-set on finding the roots, I'm probably not the guy to answer the question, but maybe there are some numerical methods that would be satisfying to you.

Answer 2

Unless you have reason to believe otherwise, it is likely that this polynomial cannot be factored algebraically. However, you can gain some insight into its roots, including the following (you're welcome to prove some or all of these claims yourself):

1. Since the coefficients of the polynomial are all positive, its real roots can only be negative.

2. Since its derivative is strictly positive over the real numbers, it is strictly increasing and therefore only has one real root.

3. As a result, its other 4 roots must come as two complex conjugate pairs, i.e. the roots can be written as $a_1, a_2 \pm i b_2, a_3 \pm i b_3$.

4. Using Vieta's formulas, we can produce some relationships between the roots, starting with $a_1 + a_2 + a_3 = -\frac{1}{3}$.