MathematicsExam code: 0580
Expanding and Factorising Brackets
Algebra and Sequences
| The expression | Method |
|---|---|
| Every term shares a factor | Take out the common factor first |
| Two square terms subtracted | Difference of two squares |
| Three terms, x² coefficient 1 | Find a pair multiplying to c, adding to b |
| Three terms, x² coefficient not 1 | Find a pair multiplying to ac, adding to b, then group |
| Four terms | Factorise in pairs and look for a common bracket |
- Always take out a common factor before anything else, since it makes what remains simpler
- A quadratic factorises with integers only when a pair multiplying to ac and adding to b exists, so a short search settles it
Worked example
Choosing the method
Factorise each expression: (a) 6x² + 9x (b) x² − 36 (c) x² − 9x + 14
Solution:
- (a) Two terms, and both share a factor. 3x divides both, so take it out
- 6x² + 9x = 3x(2x + 3)
- (b) Two terms, one square subtracted from another, and no common factor
- That is the difference of two squares: x² − 6² = (x − 6)(x + 6)
- (c) Three terms with no number in front of x², so look for two numbers multiplying to +14 and adding to −9
- Both must be negative to give a positive product and a negative sum: −2 and −7
- x² − 9x + 14 = (x − 2)(x − 7)
- Always test for a common factor first, because taking one out can turn what is left into one of the standard forms