# How do you find the roots of a cubic equation?

Solving cubics is an interesting problem: while there is a formula which can find the roots of every cubic equation, it isn't taught and is not generally worth learning. Instead, exam questions will often give you a root of a cubic, and from that you are expected to fully factorise it, and hence find the roots. Let's look at an example!

Q: Given that -2 is a root of 2x^3 + 9x^2 - 2x - 24, find all roots.

A: Firstly, we know by the factor theorem that if ** a** is a root of a polynomial (a cubic, for instance), then

**will be a factor of that polynomial. Therefore, we know that**

*(x - a)**is a factor of*

**(x + 2)***. To get the other roots, we could use polynomial division, but there is a way which is quicker and less error-prone. Write this as such:*

**2x^3 + 9x^2 - 2x - 24****2x^3 + 9x^2 - 2x - 24 = (x + 2)( )**

Now, we know that in the brackets there will be an * x^2* term, an

*term and a constant. What can the*

**x***term be? Well it must be*

**x^2***, because when we multiply out the brackets, we need to end up with*

**2x^2***, and the*

**2x^3****only**way we get a cubic term here is by multiplying the

**by some**

*x***term.**

*x^2***2x^3 + 9x^2 - 2x - 24 = (x + 2)(2x^2 )**

Similarly, the constant term must be -12, because we need a -24 after multiplying out the brackets, and the **only** way to get a constant term here is by multiplying the two constant terms.

**2x^3 + 9x^2 - 2x - 24 = (x + 2)(2x^2 - 12)**

Now the ** x** term. if we start to multiply out, we see that we have

*. We have*

**2x^3 + 4x^2 - 12x - 24***, which we got from multiplying*

**4x^2****by**

*2***, but we need**

*2x^2**, so we have to add on 5 more. The other way to get an*

**9x^2****term is to multiply two**

*x^2**terms. So our*

**x****term must be**

*x***, so that when we multiply it by the**

*5x***in the**

*x**, we end up with the extra*

**(x + 2)****.**

*5x^2***2x^3 + 9x^2 - 2x - 24 = (x + 2)(2x^2 + 5x - 12)**

Finally, we just have to factorise the quadratic in the bracket. Using inspection, or failing that the quadratic formula (though this is more prone to error), we find that:

**2x^3 + 9x^2 - 2x - 24 = (x + 2)(2x - 3)(x + 4)**

Applying the factor theorem again, we find that the roots are -4, -2 and 3/2.