Algebra · Algebraic manipulation
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Factorising with common factors
Why factorising is expansion run backwards, what "factorise fully" demands, and why a difference of two squares splits instantly — plus what the factorised form is actually FOR.
Algebra · Algebraic manipulation
Factorising with common factors
Why factorising is expansion run backwards, what "factorise fully" demands, and why a difference of two squares splits instantly — plus what the factorised form is actually FOR.
Why it works
Factorising is expanding in reverse — rewriting a sum as a product. : both terms are divisible by 3, so pull the 3 out front and record what's left inside: . Expanding the answer must recreate the original — that check is free and catches nearly every slip."Factorise fully" means the HIGHEST common factor comes out — the biggest number that divides every coefficient, and each shared letter at its lowest power. For : numbers give 6, the 's share one , the 's share one , so
Taking out less — say — is not wrong, but it isn't full: the inside still has a common factor, and the exam mark says fully.
The factor comes OUT, it doesn't vanish. is , not — dividing the expression by changes its value. Factorised form is the same expression, re-packaged as a product.
A difference of two squares needs no factor hunt. : expanding gives cross terms that cancel, which is exactly why no middle term appears. Anything of the shape splits the same way, letters included: . But is NOT — that expands to ; a square of a bracket always carries its middle term.
Why factorise at all? A product reveals what a sum hides: factors show what an expression is divisible by. That's the engine of solving (zero-product rule), of cancelling in fractions, and of proof — e.g. looks opaque, but as a difference of two squares it's : visibly a multiple of 8 for every whole .