factoring left side to product with 3 terms

Question

I want to solve the following problem:

$2x^3 +x^2 -50x = 25$

I first subtracted both sides by 25:

$2x^3 +x^2 -50x -25 = 0$

But then I got stuck, so I put in the problem in wolframalpha to see what it recommends

Wolfram solves the problem by doing what I did and then factor the left side to a product of 4:

$(x - 5)(x + 5)(2x + 1) = 0$

My problem with this is, how am I supposed to come to that conclusion without using a computer? I seems like it would take a lot of trial and error to get the result unless there exists some algorithm I don't know of.

So my question is: What is the process of factoring $2x^3 +x^2 -50x -25 = 0$ into $(x - 5)(x + 5)(2x + 1) = 0$? Or is there another way which is easier?

Thank you!

Answer 1

Have you ever heard of the Rational Root Theorem? Basically, it says that if a polynomial has any rational roots, they will be a factor of the constant divided by a factor of the leading coefficient. There are other ways to find roots, but they usually require some calculus. There is also a closed form equation for the roots of a third degree polynomial, but no one ever uses it.

Answer 2

Notice that \begin{equation}\label{eq24} \begin{split} 2x^3 + x^2 -50x - 25 &= (2x^3 -50x) + [x^2 - 25] \\&= 2x(x^2 -25) + [x^2 - 25] \\&= (x^2 -25)(2x - 1) \end{split} \end{equation} where $x^2 - 25 = (x-5)(x+5)$