Grade 12 Maths · Calculus

Factorise a cubic polynomial

Factorise a cubic by finding a value that makes it zero, then dividing to find the quadratic factor (for example x = 1 gives the factor x − 1).

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Factorise Cubic Equation Grade 12: IntroductionKevinmathscience

Factor Theorem Grade 12: IntroductionKevinmathscience

Calculus - How to Factorise a Cubic Expression (Grade 12 Mathematics)MatheNatic

Mathematics 015: Solving cubic equations using the factor theoremSABC Education Shows

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Question 9.1 · May/June 2025 · Paper 1 · National (DBE) · 2 marks

Given:

f (x) = x^{3} - 6 x^{2} + 9 x - 4 = (x - 4) (x - k)^{2}

9.1 Show that

k = 1

.(2)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: factorise a cubic polynomial

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3. Question WE11 · Siyavula Grade 12 Mathematics, 5.4 Factor theorem, Worked example 11 · 3 marks
WE11 Factorise completely: $f (x) = x^{3} + x^{2} - 9 x - 9$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Find a factor $f (1) = (1)^{3} + (1)^{2} - 9 (1) - 9 = - 16$ , so $(x - 1)$ is not a factor.
  $f (- 1) = (- 1)^{3} + (- 1)^{2} - 9 (- 1) - 9 = 0$ , so $(x + 1)$ is a factor.
  1 mark · Factorise a cubic polynomial
2. ✓
  Quadratic factor by inspection $x^{3} + x^{2} - 9 x - 9 = (x + 1) (x^{2} + ? x - 9)$
  $(+ 1) (x^{2})$ already gives the $x^{2}$ term, so the middle coefficient is $0$ .
  $f (x) = (x + 1) (x^{2} - 9)$
  1 mark · Factorise a cubic polynomial
3. ✓
  Factorise fully $f (x) = (x + 1) (x - 3) (x + 3)$
  1 mark · Factorise a difference of squares
4. Question WE13 · Siyavula Grade 12 Mathematics, 5.5 Solving cubic equations, Worked example 13 · 4 marks
WE13 Solve: $6 x^{3} - 5 x^{2} - 17 x + 6 = 0$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Find a factorLet $f (x) = 6 x^{3} - 5 x^{2} - 17 x + 6$
  $f (1) = 6 - 5 - 17 + 6 = - 10$ , so $(x - 1)$ is not a factor.
  $f (2) = 6 (2)^{3} - 5 (2)^{2} - 17 (2) + 6 = 48 - 20 - 34 + 6 = 0$ , so $(x - 2)$ is a factor.
  1 mark · Factorise a cubic polynomial
2. ✓
  Quadratic factor $6 x^{3} - 5 x^{2} - 17 x + 6 = (x - 2) (6 x^{2} + 7 x - 3)$
  1 mark · Factorise a cubic polynomial
3. ✓
  Factorise fully $6 x^{3} - 5 x^{2} - 17 x + 6 = (x - 2) (2 x + 3) (3 x - 1)$
  1 mark · Factorise trinomials with a leading coefficient
4. ✓
  Answer $(x - 2) (2 x + 3) (3 x - 1) = 0$
  $x = 2$ or $x = \frac{1}{3}$ or $x = - \frac{3}{2}$
  1 mark · Solve a quadratic by factorising

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