Solve cubic equations from their factors
Turn a factorised cubic into a complete set of roots and check that no factor has been overlooked.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
If A(x)B(x) = 0, what must be true?
What you will learn
- Solve a cubic after obtaining its linear and quadratic factors
- Find real roots from each factor
- Check solutions in the original equation
Set every factor equal to zero
If a cubic is written as (x − a)(x − b)(x − c) = 0, the zero-product rule gives x = a, b or c. A quadratic factor may contribute two roots or none, depending on its discriminant.
If a remaining quadratic does not factorise simply, use the quadratic formula. Report only roots that satisfy the original equation and any stated domain.
Check the count and substitution
A cubic can have one or three real roots, counting repeated roots. Do not assume every cubic has three distinct real solutions.
Substitute each proposed root into the original polynomial or verify the factorisation by expansion.
Pause and explain
How many real roots does (x − 1)(x² + 1) = 0 have?
Worked example
Solve x³ − 2x² − 5x + 6 = 0, given that x = 1 is a root.
Show the worked solution
- Use the factor theorem to get (x − 1)(x² − x − 6) = 0.
- Factor the quadratic: (x − 1)(x − 3)(x + 2) = 0.
- Set each factor equal to zero.
Answer x = −2, 1 or 3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve (x − 2)(x + 1)(x − 5) = 0.
Give me a hint
Set each linear factor equal to zero.
Compare my reasoning
- x − 2 = 0 gives x = 2.
- x + 1 = 0 gives x = −1.
- x − 5 = 0 gives x = 5.
x = −1, 2 or 5.
Look for these in your work
- Every factor considered
- Signs solved correctly
- All roots reported
Solve x³ − 4x² + x + 6 = 0, given x = 2 is a root.
Give me a hint
Divide by x − 2, then solve the quadratic quotient.
Compare my reasoning
- Division gives x² − 2x − 3.
- Factor the quotient as (x − 3)(x + 1).
- Set all three factors equal to zero.
x = −1, 2 or 3.
Look for these in your work
- Quotient correct
- Quadratic factored
- All roots stated
A cubic model for a signed quantity is zero at several thresholds. Why should a solver verify the factorisation before using its roots as thresholds?
Give me a hint
A wrong factor changes both the zero locations and the model's sign intervals.
Compare my reasoning
- Expand the proposed factors or substitute each root into the original polynomial.
- Check that every candidate makes the original expression zero.
- Keep only real roots within any model domain.
Verification confirms the thresholds solve the original model, and a domain check removes impossible values.
Look for these in your work
- Original equation used
- All candidates checked
- Domain limitation recognized
Common mistakes
- Giving only the supplied root
- Losing a negative root when solving x + 2 = 0
- Using a factorisation that does not expand back to the cubic
What can you explain now?
What roots follow from (x + 4)(x² − 9) = 0?
Compare with the explanation
x = −4, −3 or 3
The linear factor gives −4 and the difference of squares gives −3 and 3.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Solve a factorised cubic by explaining the zero-product step and the final check.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.