1. Introduction
A cubic expression is an algebraic expression where the highest power of the variable is 3, such as:
\(ax^3 + bx^2 + cx + d\)
Cubic factorisation is slightly more advanced than quadratic factorisation, but once we understand the patterns, it becomes manageable. We normally use one of these methods:
- Take out common factor
- Grouping method
- Cubic identities
- Finding one factor using trial and error
2. Methods of Factorising Cubics
Based on the type of expression, different methods work better.
2.1. 1. Taking Out Common Factor
If all terms share a common factor, always take it out first.
Example:
\(5x^3 + 10x^2 + 15x = 5x(x^2 + 2x + 3)\)
2.2. 2. Factorising by Grouping
Works mostly for 4-term cubic expressions.
Example:
Factorise: \(x^3 + 3x^2 + x + 3\)
Group:
\((x^3 + 3x^2) + (x + 3)\)
Common factors:
- \(x^2(x + 3)\)
- \(1(x + 3)\)
Final:
\((x + 3)(x^2 + 1)\)
2.3. 3. Using Cube Identities
If the expression matches a cube identity, use it directly.
Identity (Sum of Cubes): \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Identity (Difference of Cubes): \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
2.4. 4. Finding One Factor by Trial
For random cubic expressions, substitute simple values like \(x = 1, -1, 2, -2\). If it makes the expression zero, then that value gives a factor.
If \(f(k) = 0\), then \((x - k)\) is a factor.
3. Examples Using Identities
3.1. Example 1: Sum of Cubes
Factorise: \(27 + x^3\)
Write as cubes:
\(27 = 3^3\), \(x^3 = x^3\)
Apply identity:
\((3 + x)(9 - 3x + x^2)\)
3.2. Example 2: Difference of Cubes
Factorise: \(64m^3 - 1\)
Write as:
\((4m)^3 - 1^3\)
Final Answer:
\((4m - 1)(16m^2 + 4m + 1)\)
4. Examples Using Grouping
Example: Factorise \(x^3 - 4x^2 + 3x - 12\)
Group:
\((x^3 - 4x^2) + (3x - 12)\)
Common:
- \(x^2(x - 4)\)
- \(3(x - 4)\)
Take binomial:
\((x - 4)(x^2 + 3)\)
5. General Cubic Strategy (When No Pattern Is Visible)
For a cubic \(ax^3 + bx^2 + cx + d\):
- Check common factor.
- Try grouping.
- Try cube identity.
- Try simple roots like 1, -1, 2, -2.
- Once one factor is found, divide the cubic to get a quadratic.
- Factorise the quadratic.
6. More Mixed Examples
1. \(x^3 - 8\)
\(= x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)\)
2. \(8x^3 + 12x^2 + 6x\)
Common = \(2x\)
→ \(2x(4x^2 + 6x + 3)\)
3. \(p^3 + 4p^2 + p + 4\)
Group → \((p^3 + 4p^2) + (p + 4)\)
→ \(p^2(p + 4) + 1(p + 4)\)
Final → \((p + 4)(p^2 + 1)\)
7. Common Mistakes
- Forgetting to check for a common factor first.
- Confusing \(a^3 + b^3\) with \((a + b)^3\).
- Wrong signs in cube identities.
- Incorrect grouping of terms.
- Not verifying whether identity matches exactly.
8. Quick Practice
Factorise:
- \(x^3 + 125\)
- \(27y^3 - 1\)
- \(x^3 + x^2 + x + 1\)
- \(64 - x^3\)
9. Summary
- Cubics can be factorised using common factors, grouping, identities, or trial methods.
- Important identities: \(a^3 + b^3\) and \(a^3 - b^3\).
- Grouping works very well for 4-term expressions.
- Always verify the factorisation by expanding.