PLP Mathematics Module 2 Factorisation

Module 2 Unit 5 FACTORISATION Downloaded from hlayiso.com

When you have completed this unit, you will be able to:  1. Determine the HCF of an algebraic expression  2. Factorise polynomials by finding the common factor 5.1 The HCF of an algebraic expression In Module 1, you learnt how to determine the Highest Common Factor (HCF) of a group of numbers. Just to refresh your memory: 1. Step 1: write all the numbers as products of their prime factors 2. Step 2: pick out ONLY the factors that appear in every number 3. Step 3: Write down those factors at their LOWEST power This will give you the HCF of the numbers. Downloaded from hlayiso.com

In algebra, there will be variables too, but you just carry on following the same steps as before. If there are NO common factors, the 𝐻𝐻𝐶𝐶𝐹𝐹 = 1. Example: Find the HCF of the following algebraic expressions: 3𝑝𝑝; 6𝑝𝑝2 𝑞𝑞; 18𝑝𝑝𝑞𝑞2 Answer: Factorise: 3𝑝𝑝 = 3 × 𝑝𝑝 6𝑝𝑝2 𝑞𝑞 = 2 × 3 × 𝑝𝑝2 × 𝑞𝑞 18𝑝𝑝𝑞𝑞2 = 2 × 32 × 𝑝𝑝 × 𝑞𝑞2 The only number that is in every expression is 3, which I will use at its lowest power and the only variable that is in every expression is 𝑝𝑝, which I will use at its lowest power. Downloaded from hlayiso.com Final answer: 𝐻𝐻𝐶𝐶𝐹𝐹 = 3𝑝𝑝

5.2 Factorise polynomials by finding the common factor In mathematics, you will often be instructed to “factorise”. What is the purpose of factorising? Factorisation is a way of changing a polynomial into a monomial, in other words, to get rid of " + " and " − " signs that separate the terms in a polynomial and replace them with " × " signs that do not separate terms. Why do we need to be able to do that? Because the Laws of Exponents and Logarithmic Laws and many other kinds of calculations in mathematics are only possible after you have gotten rid of " + " and " − " signs by factorising. I am going to use ordinary numbers first just to show you what I mean. Let us say that I want to do the following: 12 + 32, I can work out the answer, which is 44, but I used a " + " in my calculation. Downloaded from hlayiso.com If I find the HCF of 12 and 32, it will be 22 = 4.

We are now going to factorise by taking out the HCF. If you take out the HCF, you write it in front of a set of brackets. Then inside the brackets, you put the numbers you need to give you the same values that you had in the beginning. 12 + 32 𝐻𝐻𝐶𝐶𝐹𝐹 = 4 = 4(3 + 8) I need 3 and 8 inside the bracket to give me 4 × 3 = 12 and 4 × 8 = 32 again. Always test that removing the bracket will give you the same expression that you started with 4 × 11 = 44 If I use BODMAS and add the 3 + 8 = 11 and then multiply, I also get 44, but now by multiplication and not by subtracting or adding. I have used factorisation to get rid of the " Downloaded + " sign. from hlayiso.com

Now we will do some examples with algebraic expressions. Example 1: Factorise the following expression: 2𝑥𝑥𝑦𝑦 + 4𝑥𝑥𝑧𝑧 Answer: Factorise the terms: 2𝑥𝑥𝑦𝑦 = 2 × 𝑥𝑥 × 𝑦𝑦 4𝑥𝑥z = 22 × 𝑥𝑥 × 𝑧𝑧 The highest common factor will be 𝟐𝟐𝒙𝒙. (I am using factors that appear in all terms, at their lowest power). 2𝑥𝑥𝑦𝑦 + 4𝑥𝑥𝑧𝑧 = 2𝑥𝑥(𝑦𝑦 + 2𝑧𝑧) Downloaded from hlayiso.com

Example 2: Factorise the following expression: 4𝑎𝑎2 𝑥𝑥𝑦𝑦 + 8𝑎𝑎𝑥𝑥 2 𝑦𝑦 + 12𝑎𝑎𝑥𝑥𝑦𝑦 2 Answer: Factorise the terms 4𝑎𝑎2 𝑥𝑥𝑦𝑦 = 22 × 𝑎𝑎2 × 𝑥𝑥 × 𝑦𝑦 8𝑎𝑎𝑥𝑥 2 𝑦𝑦 = 23 × 𝑎𝑎 × 𝑥𝑥 2 × 𝑦𝑦 12𝑎𝑎𝑥𝑥𝑦𝑦 2 = 3 × 22 × 𝑎𝑎 × 𝑥𝑥 × 𝑦𝑦 2 Decide on the HCF: it will include only factors that appear in every term and used at its lowest power: 𝐻𝐻𝐶𝐶𝐹𝐹 = 22 × 𝑎𝑎 × 𝑥𝑥 × 𝑦𝑦 = 4𝑎𝑎𝑥𝑥𝑦𝑦 Final answer: 4𝑎𝑎2 𝑥𝑥𝑦𝑦 + 8𝑎𝑎𝑥𝑥 2 𝑦𝑦 + 12𝑎𝑎𝑥𝑥𝑦𝑦 2 = 4𝑎𝑎𝑥𝑥𝑦𝑦(𝑎𝑎 + 2𝑥𝑥 + 3𝑦𝑦) Downloaded from hlayiso.com

4𝑎𝑎2 𝑥𝑥𝑦𝑦 + 8𝑎𝑎𝑥𝑥 2 𝑦𝑦 + 12𝑎𝑎𝑥𝑥𝑦𝑦 2 = 4𝑎𝑎𝑥𝑥𝑦𝑦(𝑎𝑎 + 2𝑥𝑥 + 3𝑦𝑦) If you are answering a test or doing homework, you do not have to write down all the steps. The two steps shown above are enough. Just always, make sure that if you remove the bracket, you again get the original terms that were in the question. NB: That method is called the FOIL method. (First, Outer, Inner, Last) Downloaded from hlayiso.com

Exercise 5.1 For each of the following groups of numbers, find the highest common factor (HCF): 1. 4𝑎𝑎𝑐𝑐 2 ; 8𝑏𝑏𝑐𝑐; 4𝑏𝑏 2 𝑐𝑐 2. 3𝑥𝑥 2 ; 𝑥𝑥𝑦𝑦; 9𝑦𝑦 2 3. 25𝑎𝑎3 𝑏𝑏 2 𝑐𝑐; 15𝑎𝑎2 𝑏𝑏 2 𝑐𝑐; 20𝑎𝑎2 𝑏𝑏 4 4. 12𝑐𝑐𝑑𝑑; 16𝑐𝑐 2 ; 10𝑐𝑐𝑑𝑑2 Downloaded from hlayiso.com

Corrections: 1. 4𝑎𝑎𝑐𝑐2; 8𝑏𝑏𝑐𝑐; 4𝑏𝑏2𝑐𝑐 4𝑎𝑎𝑐𝑐 2 = 22 × 𝑎𝑎 × 𝑐𝑐 2 8𝑏𝑏𝑐𝑐 = 23 × 𝑏𝑏 × 𝑐𝑐 4𝑏𝑏 2 𝑐𝑐 = 22 × 𝑏𝑏 2 × 𝑐𝑐 𝐻𝐻𝐶𝐶𝐹𝐹 = 22 𝑐𝑐 = 4𝑐𝑐 2. 3𝑥𝑥 2 ; 𝑥𝑥𝑦𝑦; 9𝑦𝑦 2 3𝑥𝑥 2 = 3 × 𝑥𝑥 2 𝑥𝑥𝑦𝑦 = 𝑥𝑥 × 𝑦𝑦 9𝑦𝑦 2 = 32 × 𝑦𝑦 2 𝐻𝐻𝐶𝐶𝐹𝐹 = 1 Downloaded from hlayiso.com

3. 25𝑎𝑎3 𝑏𝑏 2 𝑐𝑐; 15𝑎𝑎2 𝑏𝑏 2 𝑐𝑐; 20𝑎𝑎2 𝑏𝑏 4 25𝑎𝑎3 𝑏𝑏 2 𝑐𝑐 = 52 × 𝑎𝑎3 × 𝑏𝑏 2 × 𝑐𝑐 15𝑎𝑎2 𝑏𝑏 2 𝑐𝑐 = 3 × 5 × 𝑎𝑎2 × 𝑏𝑏 2 × 𝑐𝑐 20𝑎𝑎2 𝑏𝑏 4 = 22 × 5 × 𝑎𝑎2 × 𝑏𝑏 4 𝐻𝐻𝐶𝐶𝐹𝐹 = 5𝑎𝑎2 𝑏𝑏 2 4. 12𝑐𝑐𝑑𝑑; 16𝑐𝑐 2 ; 10𝑐𝑐𝑑𝑑2 12𝑐𝑐𝑑𝑑 = 22 × 3 × 𝑐𝑐 × 𝑑𝑑 16𝑐𝑐 2 = 24 × 𝑐𝑐 2 10𝑐𝑐𝑑𝑑 2 = 2 × 5 × 𝑐𝑐 × 𝑑𝑑 2 𝐻𝐻𝐶𝐶𝐹𝐹 = 2𝑐𝑐 Downloaded from hlayiso.com