0580

Expanding and Factorising Brackets

Algebra and Sequences

Every term times every term

  • 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

Squaring a bracket

  • 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
Exam tip

Multiply by the sign as well as the number. In (2x − 3)(x − 4) the second bracket is multiplied by −3, not 3, which is where the constant term goes wrong far more often than the x term does. Check the constant on its own at the end: −3 × −4 = +12.

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