How many roots can a cubic have?

A cubic always has three roots counted with repeats, but they need not all be real. It always has at least one real root, because its graph must cross the x-axis.

What do I do if no whole number is a root?

Try fractions p/q, where p divides the constant term and q divides the leading coefficient. If none works, the real roots are irrational and need a numerical method.

Cubic and Polynomial Equation Solver with Steps

Type an equation's left-hand side, such as x^3 - 2x^2 - 5x + 6, and the solver finds every root of it $= 0$ . Try to find one root yourself first.

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How to solve a cubic equation by hand

Take $x^{3} - 2 x^{2} - 5 x + 6 = 0$ .

Find one root. $p (1) = 1 - 2 - 5 + 6 = 0$ , so $x = 1$ is a root and $(x - 1)$ is a factor.
Divide it out. $x^{3} - 2 x^{2} - 5 x + 6 = (x - 1) (x^{2} - x - 6)$ .
Solve the quadratic. $x^{2} - x - 6 = (x - 3) (x + 2)$ .

x = 1, x = 3 or x = - 2

If the quadratic in step 3 does not factorise, use the quadratic equation solver or the formula.

Practise with real exam questions

O Level 4037 factor and remainder theorem: find the unknown coefficients, then solve the cubic.

Solve cubic and polynomial equations, step by step

How to solve a cubic equation by hand

Practise with real exam questions

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