SECONDARY MATHEMATICS · ALGEBRA AS STRUCTURE
Factorisation is not a collection of unrelated tricks. At its core, it reverses multiplication over addition and turns a sum into a product without changing the expression’s value.
Start from distribution
The distributive law says a(b + c) = ab + ac. Read from left to right, it expands a product. Read from right to left, it factorises a sum: ab + ac = a(b + c).
For example, 6x + 9 = 3(2x + 3). Expanding the right side returns 6x + 9. Factorisation changes the structure while preserving value.
Common factor first
In 12x² − 18x, both terms contain 6x. Factorising gives 6x(2x − 3). The factor 6x is not guessed; it is a quantity that multiplies each term.
A useful check is to expand: 6x·2x − 6x·3 = 12x² − 18x. If expansion does not return the original expression, the factorisation is incorrect.
Quadratics reveal a hidden product
Consider x² + 7x + 12. We seek two numbers whose product is 12 and whose sum is 7: 3 and 4. Therefore x² + 7x + 12 = (x + 3)(x + 4).
Why? Expanding (x + 3)(x + 4) gives x² + 4x + 3x + 12 = x² + 7x + 12. The familiar “sum and product” search is a compressed way of reversing this expansion pattern.
Difference of two squares
(a − b)(a + b) expands to a² + ab − ab − b² = a² − b². The middle terms cancel. Therefore a² − b² = (a − b)(a + b).
So x² − 25 = (x − 5)(x + 5). The pattern works because 25 = 5² and the expression is a difference. x² + 25 does not factor over the real numbers using this identity.
Perfect-square structure
(x + a)² = x² + 2ax + a². Therefore x² + 10x + 25 = (x + 5)². The middle coefficient 10 is twice 5, and the constant 25 is 5².
Recognising this structure can make later equation solving or algebraic simplification shorter, but recognition should be checked by expansion.
Why factorisation helps solve equations
Suppose x² + 7x + 12 = 0. Factorisation gives (x + 3)(x + 4) = 0. A product is zero when at least one factor is zero, so x = −3 or x = −4.
The zero-product property is available because factorisation exposed a product. The roots were already encoded in the quadratic; the new form makes them visible.
Factorisation and cancellation
In (x² − 4)/(x − 2), factorisation gives (x − 2)(x + 2)/(x − 2). For x ≠ 2, the common nonzero factor cancels and the expression becomes x + 2. The original restriction x ≠ 2 remains.
Factorisation can therefore expose a common factor, but it does not erase domain conditions. This connects to equivalent expressions.
A factorisation decision tree
- Is there a common factor in every term? Extract it first.
- Is the remaining expression a difference of two squares?
- Is it a quadratic that may split into two linear factors?
- Does it match a perfect-square pattern?
- Expand the proposed factors to verify the result.
Worked practice
A. 8x + 12 = 4(2x + 3).
B. x² + 9x + 20 = (x + 4)(x + 5).
C. 9y² − 16 = (3y − 4)(3y + 4).
D. x² − 12x + 36 = (x − 6)².
E. 3x² + 12x = 3x(x + 4).
The deeper idea
Expansion and factorisation are not competing chapters. They are opposite directions through the same distributive structure. A student who understands that relationship can reconstruct many factorisation patterns instead of depending only on memorised templates.
Continue through Algebraic Structure and the substitution-checking article, then return to the Secondary Mathematics Master Index.