Dissolution of a polynomial

Suppose you have a polynomial of, for example, degree 3:

Because the first term is 5, you can imagine that it had 3 roots like:

Then to find a,b, and c solve for what terms multiply to if you expanded the above:

It gives:

Plugging back in:

Multiplying the above out gives something very close to the original polynomial (excepting rounding error)

Now you can find where those 3 terms equal 0 to find the roots, so it changes finding the roots into a problem about solving a set of simultaneous equations. Though if all you are doing is setting the polynomial to 0, it would be easier to divide out the leading 5 coefficient and simplify the formula...