Solving Cubic Equations (Cambridge O Level Additional Maths)

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,