Factorization • Topic 3 of 3

Factorisation of Cubic Polynomials

How to factorize cubic polynomials? A cubic polynomial has the form p(x) = \(ax^{3} + bx^{2} + cx + d\), where a ≠ 0. To factorize it completely, we follow these steps:

Step 1: Find the first factor

  • Use the Factor Theorem to find a value 'a' such that p(a) = 0
  • Try possible factors: ±(factors of d)/(factors of a) (Rational Root Theorem)
  • Common test values: ±1, ±2, ±3, ±1/2, etc.

Step 2: Perform polynomial division

  • Divide p(x) by (x − a) to get a quadratic quotient

Step 3: Factorize the quadratic

  • Factorize the quadratic expression using middle term splitting or quadratic formula

Step 4: Write the complete factorization

  • p(x) = (x − a) × (quadratic factors)

Rational Root Theorem (for finding possible zeros): If p(x) = \(ax^{3} + bx^{2} + cx + d\) has integer coefficients, then any rational zero is of the form: p/q where p divides the constant term d and q divides the leading coefficient a

Step pipeline to factorise a cubic: find a zero, divide out the linear factor, factorise the quadratic Factorising a cubic — the steps 1. Find a zero a (Rational Root: factors of d / factors of leading coeff) 2. (x − a) is a factor — divide p(x) by (x − a) 3. Factorise the quadratic quotient by splitting the middle term 4. p(x) = (x − a) × (two linear factors) Factor tree splitting x cubed minus 6x squared plus 11x minus 6 into x minus 1, x minus 2 and x minus 3 Factor tree — roots 1, 2, 3 x^3 − 6x^2 + 11x − 6 (x − 1) x^2 − 5x + 6 (x − 2) (x − 3) p(x) = (x − 1)(x − 2)(x − 3)
Solved Examples
1
Worked Example
Factorize p(x) = \(x^{3} - 3x^{2} - 9x - 5\).
Solution
  1. Step 1: Try x = −1: p(−1) = −1 − 3 + 9 − 5 = 0 → (x + 1) is a factor
  2. Step 2: Divide p(x) by (x + 1):
  3. \(x^{2} - 4x - 5\)
  4. ──────────────
  5. x+1 ) \(x^{3} - 3x^{2} - 9x - 5\)
  6. \(x^{3} + x^{2}\)
  7. ─────────
  8. −\(4x^{2} - 9x
  9. 4x^{2} - 4x\)
  10. ──────────
  11. −5x − 5
  12. −5x − 5
  13. ───────
  14. 0
  15. Step 3: Factorize \(x^{2} - 4x - 5\) = (x − 5)(x + 1)
  16. Step 4: p(x) = (x + 1)(x − 5)(x + 1) = (x + 1)\(^{2}\)(x − 5)

Answer: (x + 1)\(^{2}\)(x − 5)

2
Worked Example
Factorize p(x) = \(2x^{3} - 3x^{2} - 11x + 6\).
Solution
  1. Step 1: Try x = 2: p(2) = 16 − 12 − 22 + 6 = −12 ≠ 0
  2. Step 2: Try x = −2: p(−2) = −16 − 12 + 22 + 6 = 0 → (x + 2) is a factor
  3. Step 3: Divide by (x + 2):
  4. \(2x^{2} - 7x + 3\)
  5. ───────────────
  6. x+2 ) \(2x^{3} - 3x^{2} - 11x + 6\)
  7. \(2x^{3} + 4x^{2}\)
  8. ──────────
  9. −\(7x^{2} - 11x
  10. 7x^{2} - 14x\)
  11. ───────────
  12. 3x + 6
  13. 3x + 6
  14. ──────
  15. 0
  16. Step 4: Factorize \(2x^{2} - 7x + 3\) = \(2x^{2} - 6x - x + 3\) = 2x(x − 3) − 1(x − 3) = (x − 3)(2x − 1)

Answer: (x + 2)(x − 3)(2x − 1)

3
Worked Example
Factorize p(x) = \(6x^{3} - 5x^{2} - 17x + 6\).
Solution
  1. Step 1: Try x = 1: p(1) = 6 − 5 − 17 + 6 = −10 ≠ 0
  2. Step 2: Try x = −1: p(−1) = −6 − 5 + 17 + 6 = 12 ≠ 0
  3. Step 3: Try x = 2: p(2) = 48 − 20 − 34 + 6 = 0 → (x − 2) is a factor
  4. Step 4: Divide by (x − 2):
  5. \(6x^{2} + 7x - 3\)
  6. ──────────────
  7. x−2 ) \(6x^{3} - 5x^{2} - 17x + 6\)
  8. \(6x^{3} - 12x^{2}\)
  9. ───────────
  10. \(7x^{2} - 17x\)
  11. \(7x^{2} - 14x\)
  12. ──────────
  13. −3x + 6
  14. −3x + 6
  15. ───────
  16. 0
  17. Step 5: Factorize \(6x^{2} + 7x - 3\) = \(6x^{2} + 9x - 2x - 3\) = 3x(2x + 3) − 1(2x + 3) = (2x + 3)(3x − 1)

Answer: (x − 2)(2x + 3)(3x − 1)

Key Points

  • Cubic polynomials have degree 3 and at most 3 linear factors
  • Use Factor Theorem to find the first factor by trial
  • Rational Root Theorem helps find possible zeros: ±(factors of d)/(factors of a)
  • After finding one factor, divide to get a quadratic
  • Factorize the quadratic by middle term splitting
  • Check your answer by multiplying the factors back
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.To factorise a cubic, first find one root using the:
Explanation: Find a factor, then divide.
Q2.A cubic polynomial has at most how many linear factors?
Explanation: Degree $3$.
Q3.$x^3-1$ factorises as:
Explanation: Difference of cubes.
Q4.After finding one linear factor of a cubic, the remaining factor is a:
Explanation: Divide to get a quadratic.
Practice more MCQs → 📝 Assignment · 35 marks