MathematicsExam code: 0580

Expanding and Factorising Brackets

Algebra and Sequences

Two brackets of two terms need four multiplications
Work through the first bracket's terms in turn, multiplying each by the whole second bracket
Collect like terms afterwards, which usually leaves three terms
- (x + 6)(x − 2) gives x² − 2x + 6x − 12, which collects to x² + 4x − 12
A grid keeps track of the products when the coefficients are awkward

A bracket squared is that bracket multiplied by itself, so it expands as a pair of brackets like any other
- (x + 3)² is (x + 3)(x + 3), which expands to x² + 6x + 9
Squaring the two terms separately is the error this guards against: x² + 9 is not the expansion, because the two middle products are missing
The middle term is always twice the product of the two terms in the bracket
- (2x − 5)² gives 4x² − 20x + 25

Worked example

Expanding a pair of brackets

Expand and simplify (3x − 4)(2x + 5).

Solution:

Every term in the first bracket must multiply every term in the second, which gives four products from two pairs of brackets
Take the terms with their signs: the first bracket holds +3x and −4
Multiply 3x by both terms of the second bracket: 3x × 2x = 6x², and 3x × 5 = 15x
Multiply −4 by both terms: −4 × 2x = −8x, and −4 × 5 = −20
Writing all four before collecting is worth a mark on its own: 6x² + 15x − 8x − 20
Only the two x terms are like terms, so only they combine: 15x − 8x = 7x
The answer is 6x² + 7x − 20
Check with a value: x = 1 gives (−1)(7) = −7 in the question, and 6 + 7 − 20 = −7 in the answer

Worked example

Two brackets with two letters

Expand and simplify (2x − 3y)(x + 4y).

Solution:

Treat the second letter exactly like a number: there are still four products, and each takes the signs of both its terms
2x × x = 2x², and 2x × 4y = 8xy
−3y × x = −3xy, and −3y × 4y = −12y²
8xy and −3xy are like terms, because yx and xy are the same, so they collect to 5xy
The answer is 2x² + 5xy − 12y²
Check with values: x = 2 and y = 1 give 1 × 6 = 6 in the question, and 8 + 10 − 12 = 6 in the answer

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