How to factorise a cubic without a calculator

Every cubic factorises the same way: find one root, use it to split off a factor, then deal with the quadratic that is left.

The method

1. Find a root. Try the factors of the constant term: $\pm 1,\pm 2,\pm 3,\dots$
2. Use the factor theorem. If $p(a)=0$, then $(x-a)$ is a factor.
3. Divide the cubic by $(x-a)$ to get a quadratic.
4. Factorise the quadratic.

Worked example

Factorise $p(x)=2x^3+3x^2-8x+3$.

Step 1. Try $x=1$:

Step 2. So $(x-1)$ is a factor.

Step 3. Divide. You can do it by long division, synthetic division, or by matching coefficients:

Step 4. Factorise the quadratic. Two numbers that multiply to $2 \times (-3) = -6$andaddto$5$:$6$and$-1$.

So:

The roots are $x=1$, $x=\frac{1}{2}$ and $x=-3$. Notice $\frac{1}{2}$ is a fraction: that is why you also try fractions when the leading coefficient is not 1.

Try it: hunt for the root yourself

Test candidates, watch the remainder, and find the one that gives zero.

Mistakes that cost marks

1. Sign slip in the factor. $p(-3)=0$ gives the factor $(x+3)$, not $(x-3)$.
2. Stopping at one factor. "Factorise completely" means the quadratic must be split too.
3. Only testing whole numbers when the leading coefficient is not 1.

More practice: 4037 factor and remainder theorem questions, or check any cubic with the polynomial factoriser.