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Factorising and solving cubic equations Cambridge IGCSE Additional Mathematics revision

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In plain words

A cubic can have up to three solutions. Finding them takes two steps: hunt for one root by trying small numbers, then divide it out. What is left is a quadratic, and you already know how to deal with those.

5 things to know

  1. Step 1: try small whole numbers, such as 1, −1, 2 and −2, in the cubic until one gives 0. That number is a root, and it gives a linear factor.
  2. Step 2: divide the cubic by that linear factor to get a quadratic. This can be done by long division or by comparing coefficients.
  3. Step 3: factorise the quadratic, or solve it with the formula.
  4. The roots to try are factors of the constant term.
  5. A cubic equation has at most three real roots.

Worked example

Solve x³ − 6x² + 11x − 6 = 0.

  1. Try x = 1: 1 − 6 + 11 − 6 = 0. So (x − 1) is a factor.
  2. Divide: x³ − 6x² + 11x − 6 = (x − 1)(x² − 5x + 6).
  3. Factorise the quadratic: (x − 1)(x − 2)(x − 3) = 0.
  4. x = 1, x = 2 or x = 3.

Worked example

Factorise 2x³ + 3x² − 11x − 6 completely.

  1. Try x = 2: 16 + 12 − 22 − 6 = 0. So (x − 2) is a factor.
  2. The quadratic factor starts with 2x² and ends with +3, since −2 × 3 = −6: (x − 2)(2x² + kx + 3). Matching the x² terms, k − 4 = 3, so k = 7.
  3. 2x² + 7x + 3 = (2x + 1)(x + 3). The answer is (x − 2)(2x + 1)(x + 3).

Tips and tricks

  • Show the substitution that finds the first factor. An answer with no method may not earn full marks on the non-calculator paper.
  • Check the constant: the last terms of the three brackets must multiply to give the constant of the cubic.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Factorising and solving cubic equations: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What are the roots of (x − 1)(x + 2)(x − 4) = 0?
    • −1, 2 and −4
    • 1, −2 and 4 (the answer)
    • 1, 2 and 4
    • −1, −2 and −4

    Each bracket is set equal to 0.

  2. What is the first step in factorising a cubic?
    • differentiate it
    • find a value of x that makes it equal to 0 (the answer)
    • complete the square
    • take the cube root

    That value gives a linear factor.

  3. What are the solutions of x³ − x = 0?
    • 0 and 1
    • 1 and −1
    • 0, 1 and −1 (the answer)
    • 0 only

    x(x − 1)(x + 1) = 0.

  4. x³ − 6x² + 11x − 6 = (x − 1) × a quadratic. What is the quadratic?
    • x² + 5x + 6
    • x² − 5x + 6 (the answer)
    • x² − 6x + 6
    • x² − 5x − 6

    Multiply back to check: the constant is −1 × 6 = −6.

  5. What is the greatest number of real roots a cubic equation can have?
    • 1
    • 2
    • 3 (the answer)
    • 4

    One for each linear factor.

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