Expanding Brackets (Cambridge (CIE) A Level Maths): Revision Note

Expanding Quadratics

You should be familiar with all aspects of basic algebra including manipulating expressions; simplifying; expanding and factorising brackets with both linear and quadratics.  In Pure Mathematics it is important that you are confident working with quadratics.

How do we expand two brackets?

• You will have learnt a method for expanding double brackets in previous courses

  • The quickest and most common method is to use FOIL

    • First term in each bracket

    • Outside terms

    • Inside terms

    • Last terms

  • For (ax + b)(cx + d) you will have acx² + adx + bcx + bd

• Once you have expanded and multiplied all four terms simplify the two middle terms to give you a three-term quadratic

  • acx² + (ad + bc)x + bd

• Take extra care when working with negatives

• Look out for a linear expression squared, this must still be treated as double brackets

  • (ax + b)² = a²x² + 2abx + b²

• Makes sure you can recognise when a problem involves the difference of two squares

  • (ax + b)(ax – b) = a²x² + abx - abx - b² = a²x² - b²

  • Spotting this will be particularly useful in many areas of Pure Mathematics

What if there is a quadratic in one (or both) of the brackets?

• Treat these the same as expanding double brackets with two linear expressions

• Take care with powers, remember the laws of indices

• Look carefully to see if the terms can be simplified or not

  • For (ax² + b)(cx² + d) you will have acx⁴ + adx² + bcx² + bd = acx⁴ + (ad + bc)x² + bd

  • For (ax² + b)(cx + d) you will have acx³ + adx² + bcx + bd

Further Expanding

What do I need to know about expanding brackets?

• To expand brackets, the rule is that each term in one set of brackets must be multiplied by each term in the other set of brackets

• 'FOIL' is a special case of this, when each set of brackets contains two terms

• If you are trying to expand something like (a + b)ⁿ for powers of n greater than 2 or 3, use the binomial expansion

• If you have to expand more than two sets of brackets, just expand them two at a time: