Solving Cubic Equations (Edexcel IGCSE Further Pure Maths)

Solving Cubic Equations

How many real solutions can a cubic equation have?

• A cubic equation 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 , and can be written as a product of three linear factors

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

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

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    • The quadratic factor will not have any real roots

    • So its discriminant  will be negative

How do I solve cubic equations?

• Suppose you have a cubic equation

• 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  by substituting values into the equation until it equals 0

    • Try small integer values ()

• If you know a root then you know a factor

  • This is because of the factor theorem

    • If is a root, then is a factor

• You can then divide by to find a quadratic factor

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

• Then you can find any other roots by solving  

  • If that equation has no real solutions, then  is the only real solution of the cubic