1. Introduction
Some expressions match well-known algebraic identities. When an expression fits one of these patterns, we can factorise it quickly and neatly using that identity.
This method is especially useful for perfect squares and cube expressions.
2. Important Algebraic Identities
Before factorising, remember these standard identities:
2.1. 1. Square Identities
Identity 1: \( (a + b)^2 = a^2 + 2ab + b^2 \)
Identity 2: \( (a - b)^2 = a^2 - 2ab + b^2 \)
Identity 3 (Difference of Squares): \( a^2 - b^2 = (a - b)(a + b) \)
2.2. 2. Cube Identities
Identity 4 (Sum of Cubes): \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)
Identity 5 (Difference of Cubes): \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)
3. How to Recognise Identity Patterns
To factorise using identities, look for:
- Perfect square expressions like \(x^2 + 6x + 9\)
- Difference of squares like \(a^2 - 16\)
- Sum/difference of cubes like \(8x^3 + 27y^3\)
- Expressions where terms match identity patterns when rearranged
4. Using Identities to Factorise
Once you see a pattern, apply the appropriate identity:
4.1. Example 1: Perfect Square Trinomial
Factorise: \(x^2 + 10x + 25\)
Check pattern:
\(x^2 + 10x + 25 = x^2 + 2·5·x + 5^2 \rightarrow (x + 5)^2\)
4.2. Example 2: Difference of Squares
Factorise: \(49 - y^2\)
Write as:
\(7^2 - y^2 = (7 - y)(7 + y)\)
4.3. Example 3: Sum of Cubes
Factorise: \(8x^3 + 27y^3\)
\(8x^3 = (2x)^3\), \(27y^3 = (3y)^3\)
Using identity:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Answer:
\((2x + 3y)(4x^2 - 6xy + 9y^2)\)
4.4. Example 4: Difference of Cubes
Factorise: \(x^3 - 125\)
\(125 = 5^3\)
\(x^3 - 5^3 = (x - 5)(x^2 + 5x + 25)\)
5. Mixed Examples
1. Factorise: \(9a^2 - 25b^2\)
This is difference of squares → \((3a - 5b)(3a + 5b)\)
2. Factorise: \(x^2 - 14x + 49\)
Perfect square → \((x - 7)^2\)
3. Factorise: \(64m^3 + 1\)
Sum of cubes → \((4m + 1)(16m^2 - 4m + 1)\)
6. Common Mistakes
- Forgetting that identities require exact pattern matching.
- Misidentifying perfect squares (e.g., 12 is not 2·something·something).
- Writing wrong signs in cube identities.
- Trying to apply identities where they do not fit.
7. Quick Practice
Factorise:
- \(x^2 - 9\)
- \(a^3 + 64\)
- \(16p^2 - 56p + 49\)
- \(125q^3 - 8\)
8. Summary
- Identities help factorise expressions instantly.
- Recognise perfect squares, sum and difference of cubes, and \(a^2 - b^2\).
- Always match patterns carefully before applying identities.