Solving Cubic Equations (Edexcel IGCSE Further Pure Maths): Revision Note

Solving Cubic Equations

How many real solutions can a cubic equation have?

• A cubic equation $a{x}^3+b{x}^2+cx+d=0$ will always have either one or three real roots (or solutions)

  • Some of these roots may be repeated

  • So it is possible to have one, two, or three unique solutions

• A cubic with three real roots $\alpha$, $\beta$ and $\gamma$ can be written as a product of three linear factors

  • $a{x}^3+b{x}^2+cx+d=\left(x-\alpha \right)\left(x-\beta \right)\left(x-\gamma \right)$
    

    • Any two of the factors could be multiplied together to give a quadratic factor

• A cubic with one real root $\alpha$ can be written as the product of a linear and a quadratic factor

  • $a{x}^3+b{x}^2+cx+d=\left(x-\alpha \right)\left(p{x}^2+qx+r\right)$

    • The quadratic factor will not have any real roots

    • So its discriminant $q^2-4pr$ will be negative

How do I solve cubic equations?

• Suppose you have a cubic equation $a{x}^3+b{x}^2+cx+d=0$

• An exam question will often give you one root

  • You may be asked to show that the root is a solution to the equation

  • Or you might have to find a root $x=\alpha$ by substituting values into the equation until it equals 0

    • Try small integer values ($1,-1,2,-2,...$)

• If you know a root then you know a factor

  • This is because of the factor theorem

    • If $x=\alpha$ is a root, then $\left(x-\alpha \right)$ is a factor

• You can then divide $a{x}^3+b{x}^2+cx+d$ by $\left(x-\alpha \right)$ to find a quadratic factor

  • $a{x}^3+b{x}^2+cx+d=\left(x-\alpha \right)\left(p{x}^2+qx+r\right)$

  • Use algebraic division, or factorise by inspection or by comparing coefficients

• Then you can find any other roots by solving  $p{x}^2+qx+r=0$

  • If that equation has no real solutions, then $x=\alpha$ is the only real solution of the cubic