Cubic Equations

Q:  Find all solutions:

x³ + x² – 11x – 3 = 0

A:  There are a few ways to do this problem, depending on what methods you have learned in class.  We first need to list the possible rational roots.  Using the rational root theorem, we know that the possibilities are ±3.

Plug in +3 for x to see if we get zero:

3³ + 3² – 11(3) – 3 = 0 ?

27 + 9 – 33 – 3 = 0 ?

0 = 0?

Yes!  Therefore, we need to use either synthetic division or polynomial long division to divide (x³ + x² – 11x – 3) by (x – 3)

(x³ + x² – 11x – 3) / (x – 3) = x² + 4x + 1

In other words, (x³ + x² – 11x – 3) factors into (x – 3)*(x² + 4x + 1):

(x³ + x² – 11x – 3) = 0

(x – 3)*(x² + 4x + 1) = 0

Since this cannot factor any more, we know that either:

x – 3 = 0  or x² + 4x + 1 = 0

x – 3 = 0 → x = 3

x² + 4x + 1 = 0 → use the quadratic equation to get that → x = -2 ± √(3)

So, final answers:

x = 3, -2 ± √(3)