Use factors to solve cubic and quartic equations completely
Check a linear factor, divide to reduce the degree and retain every distinct real root allowed by the problem.
Exam scope and assessment notes
Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Shared Algebra foundations, Mathematics 9709, 2026–2027. Paper 1 knowledge is assumed. These modulus and polynomial requirements occur in both sections 2.1 and 3.1. Paper 3 additionally requires partial fractions and rational-power binomial expansions; follow its separate lessons for those methods. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Both sections use linear modulus graphs; nonlinear y = |f(x)| and y = f(|x|) graphs are excluded. Polynomial dividends have degree at most 4 and divisors are linear or quadratic. Written practice is self-checked.
Before you begin
- Use the factor theorem and polynomial division.
- Factor and solve quadratic equations, including a difference of squares.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does f(3/2) = 0 prove about the factor 2x − 3?
What you will learn
- Verify a factor with the factor theorem and factorise the reduced quotient.
- Solve degree-three or degree-four polynomial equations and interpret repeated roots or contextual restrictions.
One known factor reduces the degree; it does not finish the solution
A linear expression ax + b with a ≠ 0 is a factor of f when f(−b/a) = 0. Verify the factor, divide f by it, then factor or solve the quotient. An equation expressed as a product equal to zero is solved by setting each factor to zero. Include roots from the original factor and from every remaining factor; a factor supplied in the question is a starting point rather than the entire root set.
For f(x) = 2x³ − 3x² − 8x + 12, f(3/2) = 0, so 2x − 3 is a factor. Division gives x² − 4 = (x − 2)(x + 2). Hence f(x) = (2x − 3)(x − 2)(x + 2), with roots 3/2, 2 and −2. Re-expanding the product and substituting the roots provide complementary checks.
Separate multiplicity, distinct roots and permitted model inputs
A quartic may factor into two quadratics or several linear factors. For x⁴ − 5x² + 4, the substitution u = x² gives u² − 5u + 4 = (u − 1)(u − 4), so the real roots are ±1 and ±2. Treat negative values of u cautiously when solving an equation in x², since real squares are non-negative. A repeated linear factor represents one distinct root with multiplicity, not multiple different inputs.
For t³ − 3t² + 4 = (t − 2)²(t + 1), the algebraic roots are t = 2, repeated, and t = −1. If a supplied time model requires t ≥ 0, only t = 2 is permitted, and it describes one distinct time. State the full algebraic root set before applying a contextual restriction so that a discarded root and its reason remain visible. Solving the equation checks the supplied model's event condition; it does not independently validate that model.
Pause and explain
For (t − 2)²(t + 1) = 0 with t ≥ 0, how many distinct permitted time values are there?
Worked example
Given that 2x − 3 is a factor of 2x³ − 3x² − 8x + 12, factorise completely and solve the equation f(x) = 0.
Show the worked solution
- The divisor root is 3/2 and f(3/2) = 0, confirming the factor.
- Division gives quotient x² − 4 = (x − 2)(x + 2).
- Set each factor in (2x − 3)(x − 2)(x + 2) to zero to obtain −2, 3/2 and 2.
Answer f(x) = (2x − 3)(x − 2)(x + 2); roots x = −2, 3/2 or 2.
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.
Given x − 2 is a factor, solve x³ − 4x² + x + 6 = 0.
Give me a hint
Divide by the supplied factor, then solve the quadratic quotient as well.
Compare my reasoning
- At x = 2 the polynomial is zero; division gives quotient x² − 2x − 3.
- Factor the quotient as (x − 3)(x + 1).
- The full product (x − 2)(x − 3)(x + 1) gives roots 2, 3 and −1.
x = −1, 2 or 3.
Look for these in your work
- Verify the supplied factor before reducing the degree.
- Include roots from every linear factor.
Factorise x⁴ − 5x² + 4 completely over the reals and solve the equation equal to zero.
Give me a hint
Use a quadratic in x², then factor each difference of squares.
Compare my reasoning
- Set u = x² to factor u² − 5u + 4 as (u − 1)(u − 4).
- Return to x: (x² − 1)(x² − 4) = (x − 1)(x + 1)(x − 2)(x + 2).
- Every factor supplies a real root, giving −2, −1, 1 and 2.
(x − 1)(x + 1)(x − 2)(x + 2); roots −2, −1, 1 and 2.
Look for these in your work
- Return from the substituted variable to the original one.
- Retain both signs for each positive square value.
A supplied event equation is t³ − 3t² + 4 = 0, where t is numerical time in seconds with t ≥ 0. Given the root t = 2, find every permitted time and explain the repeated root.
Give me a hint
Divide by t − 2 and check whether the quotient contains that factor again.
Compare my reasoning
- Division gives t² − t − 2 = (t − 2)(t + 1), so the full expression is (t − 2)²(t + 1).
- The algebraic roots are 2, with multiplicity 2, and −1; reject −1 because t ≥ 0.
- Only the distinct time t = 2 s remains. Its repeated factor does not describe two different times.
Only t = 2 s is permitted; it is a repeated root, with negative root −1 excluded.
Look for these in your work
- Find the quotient roots, including any repeated supplied root.
- Distinguish distinct permitted times from multiplicity.
Common mistakes
- Include the root from the supplied factor as well as the quotient roots.
- A repeated factor is one distinct root, not two different event times.
- Apply model restrictions after solving the algebraic equation and explain exclusions.
What can you explain now?
Given that x − 1 is a factor of x³ − 2x² − 5x + 6, solve the equation equal to zero.
Compare with the explanation
x = −2, 1 or 3.
The quotient is x² − x − 6 = (x − 3)(x + 2), so all three linear factors supply roots.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Starting from one verified factor, explain how to find all other real roots. Separate repeated factors from distinct roots and apply any model domain last.
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.