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?acx^2?+?adx?+?bcx?+?bd
• Once you have expanded and multiplied all four terms simplify the two middle terms to give you a three-term quadratic
  • acx^2?+ (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?=?a²x^2?+ 2abx?+?b²
• Makes sure you can recognise when a problem involves the?difference of two squares
  • (ax?+?b)(ax?–?b)^?=?a²x^2?+?abx?-?abx?+?b^2?=?a²x^2?+?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?acx^4?+?adx²?+?bcx²?+?bd?=?acx^4?+ (ad?+?bc)x²?+?bd
  • For (ax²?+?b)(cx?+?d) you will have?acx^3?+?adx²?+?bcx?+?bd

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)ⁿfor 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: