Solving Cubic Equations (Cambridge (CIE) O Level Additional Maths): Revision Note

Solving Cubic Equations

What is a cubic equation?

• A cubic function is an polynomial of degree 3

  • i.e.  the highest power of  is 3 

• A cubic equation can be written in the form

  • 

• Solving a cubic equation involves factorising the cubic function first.

How do I factorise a cubic function?

• Factorising a cubic (function) combines the factor theorem with the method of polynomial division

• The example below shows the steps for factorising a cubic

  

STEP 1
Use factor theorem.
Find a value such that .

STEP 2
Use polynomial division.
Divide  by .
(It is possible to do this step 'by inspection', see the worked example below)

STEP 3
Use the result of your division to write .

STEP 4
If the quadratic can be factorised, do so.
can then be written as the product of three linear factors.
If the quadratic cannot be factorised, then the result from STEP 3 is the final factorisation.

How do I solve a cubic equation?

• A cubic equation will have either 1, 2 or 3 (real) solutions

  • (The cubic function will have either 1, 2 or 3 (real) roots)

• Once the cubic function is factorised using the four steps above, there is one more step to carry out

STEP 5 Find the solutions to the cubic equation by making each factor equal to zero

• For each linear factor, 

  •  

  • so is a solution

  • This is the factor theorem!

  • For a quadratic factor, 

    •  

    • use either the quadratic formula or completing the square (as it won't factorise)

      • this will give two of the solutions to the cubic equation

    • if there are no solutions to the quadratic equation there are no solutions other than that from the linear factor

• From the example above,

  • 

  • so the solutions to the cubic equation  are

    • and 

• Cubic equations can have equal (repeated) solutions

  • e.g.   has two (equal and real) roots,  (repeated) and  

  • e.g.   has three (equal and real) roots,