Finding Roots of a Cubic Function: Tips and Tricks

[zenith92]

Homework Statement

Actually, part of a question on modeling/control systems. I need to find the poles (or, roots) of the cubic function.

Homework Equations

x^3 - 13x + 12 = 0

The Attempt at a Solution

The rule of thumb for the course is that if you get a cubic function then there will always be a solution of x equal to -1,0 or 1. For this particular equation, I figured out that it's 1. Now I need 2 more solutions.

On my first try I got +/- sqrt(13) by factoring the equation as follows:

x(x^2-13x)+12=0

Those are wrong, I checked them. So yeah, question is if someone can help me or point me in the direction of a simple way to solve cubic functions, I have almost no experience with them (normally I'd use my calculator, not allowed for this class).

The answers in the appendix say that the other two roots are 3 and -4, if that helps.

Thanks in advance!

 

[hotvette]

Since you know the first root is 1, you can write:

(x-1)(x+a)(x+b) = x³-13x+12

From this you can tell that -1*a*b = 12, thus a*b = -12. Additionally, since the x² term is zero, you can multiply the left side and set the x² term to zero, revealing that a + b = 1. The only possibility where a*b = -12 and a + b = 1 is a = 4 and b = -3, which means x = -4 and x = 3 are roots.

 

[zenith92]

Thanks mate, actually don't know why I didn't get this immediately :)