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Academic Skills Advice Factorisation & Function Notation Factorisation: Some expressions can be simplified by factorising. This means taking out (dividing by) any common factors (i.e. numbers or letters that are in every term of the expression) and putting everything that’s left into brackets. This is the opposite to “multiplying out brackets” covered in “Basic Algebra”. A reminder of cancelling fractions: To be able to divide each term in the expression by the common factor you need to remember how to divide fractions (covered in lesson 1). Here are a few examples: 3𝑏 3𝑏 =1 4𝑥𝑦 8𝑥 = 10𝑎2 𝑏 2 5𝑎𝑏 5 (top & bottom are the same so they cancel leaving 1) 𝑦 (4 goes into 8 twice, the ‘𝑥’ at the top and bottom cancel) 2 = 2𝑎 𝑏3 (10÷5=2 and the rules of indices say that 𝑎2 ÷ 𝑎 = 𝑎2−1 = 𝑎 and 𝑏 2 ÷ 𝑏 5 = 𝑏 2−5 = 𝑏 −3 (= You could also do this one by cancelling top & bottom: 10𝑎2 𝑏 2 5𝑎𝑏 5 1 )) 𝑏3 2 10×𝑎×𝑎×𝑏×𝑏 = 5×𝑎×𝑏×𝑏×𝑏×𝑏×𝑏 = 2𝑎 𝑏3 Examples of Factorising: Simplify the following: 𝟔𝒂 + 𝟔𝒃 Both terms will divide by 6 so put 6 outside the bracket and work out what will stay inside the bracket (by dividing everything by 6). Starting expression: 6𝑎 + 6𝑏 The 6’s will cancel top & bottom 6( 6 + 6 ) So we end up with: 6(𝑎 + 𝑏) Simplify the following: 6𝑎 6𝑏 You could check this is correct by multiplying out the bracket and making sure you get back to where you started. 𝟕𝒙 + 𝟐𝟏𝒙𝒚 Both terms will divide by 7 and 𝑥 so put 7𝑥 outside the bracket and work out what will stay inside the bracket (by dividing everything by 7𝑥). Starting expression: 7𝑥 + 21𝑥𝑦 The 7𝑥 will cancel top & bottom 7𝑥 (7𝑥 + 7𝑥 ) So we end up with: 7𝑥(1 + 3𝑦) © H Jackson 2011 / ACADEMIC SKILLS 7𝑥 21𝑥𝑦 Multiply out the bracket to check it’s correct. 1 Simplify the following: Collect like terms: So we have: 𝟖𝒙 + 𝟑𝒙𝒚 + 𝟗𝒙𝒚 − 𝟒𝒙 8𝑥 − 4𝑥 = 4𝑥 3𝑥𝑦 + 9𝑥𝑦 = 12𝑥𝑦 4𝑥 + 12𝑥𝑦 Now we can factorise because each term will divide by 4 and 𝑥. So we divide every term by 4𝑥 and put it outside the bracket. Starting expression: 4𝑥 + 12𝑥𝑦 The 4𝑥 will cancel top & bottom 4𝑥 (4𝑥 + 4𝑥 ) So we end up with: 4𝑥(1 + 3𝑦) 4𝑥 Multiply out the bracket to check it’s correct. 12𝑥𝑦 Simplify the following expression: 𝟑𝒂𝒃𝟐 + 𝟒𝒂𝟐 𝒃 – 𝒃𝟐 𝒂 – 𝟓𝒂𝒃 – 𝟔𝒂𝟐 𝒃 This looks more complicated but remember you can collect any terms that are the same. Notice that 𝑎𝑏 2 is the same as 𝑏 2 𝑎 (just written the other way round) but is not the same as 𝑎2 𝑏. Collect like terms: 3𝑎𝑏 2 – 𝑏 2 𝑎 = 2𝑎𝑏 2 4𝑎2 𝑏 – 6𝑎2 𝑏 = −2𝑎2 𝑏 −5𝑎𝑏 (there is no other 𝑎𝑏 term) So we have: 2𝑎𝑏 2 − 2𝑎2 𝑏 − 5𝑎𝑏 Now we have an expression that we can factorise as each term includes an ‘𝑎’ and a ‘𝑏’ so we can divide each term by 𝑎𝑏 and put it outside the bracket. Starting expression: 2𝑎𝑏 2 − 2𝑎2 𝑏 − 5𝑎𝑏 The 𝑎𝑏 will cancel top & bottom 𝑎𝑏 ( 𝑎𝑏 − 𝑎𝑏 − 𝑎𝑏 ) So we end up with: 2𝑎𝑏 2 2𝑎2 𝑏 5𝑎𝑏 Check it’s correct. 𝑎𝑏(2𝑏 − 2𝑎 − 5) Simplify the following expression: 𝒙𝟑 𝒚𝟐 + 𝒙𝟐 𝒚𝟑 + 𝒙𝟐 𝒚 Now we need to decide on the highest powers we can divide by. Every term will divide by an ‘𝑥 2 ’ and a ‘𝑦’ so we can divide each term by 𝑥 2 𝑦 and put it outside the bracket. N.b. we cannot divide by 𝑦 2 as the last term has a lower power in (𝑦). Starting expression: 𝑥3𝑦2 + 𝑥2𝑦3 + 𝑥2𝑦 1 2 2 1 𝑥3𝑦2 2 𝑥2𝑦3 The 𝑥 𝑦 will cancel top & bottom 𝑥 𝑦 ( 𝑥2𝑦 + So we end up with: 𝑥 2 𝑦 (𝑥𝑦 + 𝑦 2 + 1) © H Jackson 2011 / ACADEMIC SKILLS 𝑥2𝑦 + 𝑥2𝑦 𝑥2𝑦 ) Check it’s correct. 2 Function Notation: Function notation is another way of writing equations. Instead of writing “𝑦 =”, we write “𝑓(𝑥) =” or “𝑔(𝑥) =” etc. We say “𝑓 of 𝑥” or “𝑔 of 𝑥”. You might see a function written as: 𝒇(𝒙) = 𝟑𝒙 − 𝟒 OR 𝒇: 𝒙 → 𝟑𝒙 − 𝟒 𝑓(𝑥) and 𝑓: 𝑥 both just mean “a function of 𝒙” and the right hand side is the algebra explaining what to do. i.e. do 3 × 𝑥 then take away 4. (Remember 𝑓(𝑥) does not mean 𝑓 × (𝑥).) Evaluating a Function: When asked to evaluate a function we just replace the 𝑥 with whatever we are finding the function of. It’s always a good idea to use brackets so that you are careful with any signs. Examples: Given the function: 𝒇(𝒙) = 𝟓𝒙 + 𝟐 Find 𝒇(𝟔) (we say “𝒇 of 6”) This just means that wherever you see an 𝑥 you replace it with 6. 𝑓(𝑥) = 5𝑥 + 2 𝑓(6) = 5(6) + 2 𝑓(6) = 5 × 6 + 2 𝑓(6) = 32 We have used 𝑓 to denote the function but we can also use other letters to mean a different function. Given the function: 𝒈(𝒙) = 𝟒𝒙𝟐 − 𝟖𝒙 Find 𝒈(−𝟑) (we say “𝒈 of – 𝟑”) Now wherever you see an 𝑥 replace it with −3. 𝑔(𝑥) = 4𝑥 2 − 8𝑥 𝑔(−3) = 4(−3)2 − 8(−3) 𝑔(−3) = 4 × (−3)2 − 8 × (−3) 𝑔(−3) = 60 Given the function: 𝒉(𝒙) = 𝟕𝒙 − 𝟓 Find 𝒉(𝟏𝟎) (we say “𝒉 of 10”) Wherever you see an 𝑥 replace it with 10. ℎ(𝑥) = 7𝑥 − 5 ℎ(10) = 7(10) − 5 ℎ(10) = 70 − 5 ℎ(10) = 65 This is a quick introduction to the notation - you will learn more about functions in lesson 10. © H Jackson 2011 / ACADEMIC SKILLS 3