Algebra
Factorisation
Break an algebraic expression into factors, and use those factors to divide one expression by another.
Expanding turns 3(x + 4) into 3x + 12. Factorising runs the film backwards: it takes 3x + 12 and finds the 3 and the (x + 4) it came from. It matters because a factorised expression is one you can cancel, solve and divide with — a sum is far harder to work with than a product.
Three methods, tried in this order
First, take out any common factor of every term. Second, look for a difference of two squares: a² - b² is always (a - b)(a + b). Third, for x² + bx + c, find two numbers that multiply to c and add to b. Always check the first method before the others — a common factor left in makes everything afterwards harder.
Worked example
Factorise 2x² - 18.
Both terms divide by 2, so take it out: 2(x² - 9).
Common factor first, always. Missing it here would leave you trying to factorise 2x² - 18 as a difference of squares, which it is not in that form.
Try it together
Now let us divide 6x² + 15x by 3x together.
Dividing algebraic expressions is factorising with one more step: factorise the top, then cancel what the bottom matches.
1.Take the highest common factor out of 6x² + 15x. What number and letter come outside the bracket?
Have a go
Have a go on your own: divide 10a³ - 4a² by 2a². Write the answer.
Hint: Divide each term of the top by 2a² in turn.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.