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 (x - a) will be a factor of that polynomial. Therefore, we know that (x + 2) is a factor of 2x^3 + 9x^2 - 2x - 24. 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 = (x + 2)(                   )

Now, we know that in the brackets there will be an x^2 term, an x term and a constant. What can the x^2 term be? Well it must be 2x^2, because when we multiply out the brackets, we need to end up with 2x^3, and the only way we get a cubic term here is by multiplying the x by some x^2 term.

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 2x^3 + 4x^2 - 12x - 24. We have 4x^2, which we got from multiplying by 2x^2, but we need 9x^2, so we have to add on 5 more. The other way to get an x^2 term is to multiply two x terms. So our x term must be 5x, so that when we multiply it by the x in the (x + 2), we end up with the extra 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.

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