Highest common factor — numerical warm-up
Find the HCF of a pair (or trio) of numbers. This is the skill you'll use to factorise.
① Single brackets — numerical common factor
Take out a numerical factor: \(6x+15=3(2x+5)\). Find the HCF of the coefficients.
② Single brackets — common factor of \(x\)
Take out a variable factor: \(x^2+5x=x(x+5)\). Every term must contain at least one \(x\).
③ Combined common factor — number and \(x\)
Take out both: \(6x^2+9x=3x(2x+3)\). Factor the numbers and the variable.
④ Difference of two squares
\(a^2-b^2=(a-b)(a+b)\). Spot the pattern: two perfect squares separated by a minus sign.
How to factorise
Step 1: Find the HCF of all the coefficients (the numbers).
Step 2: Find the highest power of \(x\) that appears in every term. Combine with step 1 to get the overall common factor.
Step 3: Divide each term by the common factor to find what's left inside the brackets.
Check: expand your answer — you should get the original expression back.
Difference of two squares: if you see \(a^2-b^2\) (a perfect square minus a perfect square), the factors are \((a-b)(a+b)\). Note that \(a^2+b^2\) does not factorise.