Unit 1: Factorising There are 4 factorising skills that you need to master at National 5 level. Pick the one you want to work on. Common Factors Difference of two squares Trinomials Completing the square Choose a button to revise a topic then use the space bar to work through the notes and examples. Hover over the buttons and text for extra information ??? What is Factorisation For ???
Common Factor Factorisations This is the easiest of the factorisation methods and it’s the one you should always check for first. The skill is in finding the largest factor that is common to every term in the expression. The factorisation pattern looks like this: ab + ac = a(b + c) The variable a multiplies both the b and the c on the left. It is called a common factor. Press the space bar to try some examples,. TIP! Always check by multiplying your answer out What can I expect in the Unit test?
Assessment style common factor questions The assessment will only test to see if you have the basic idea. Take care to write out your solutions clearly and don’t take shortcuts. Press the space bar for a selection of Unit test level questions. Write out your solutions before checking the answers. Remember that your revision needs to be active. You will learn better when you do the questions. Fully factorise these 3 expressions:
The Difference of two squares (DOTS) This factorisation method requires that you spot when a term is a perfect square. It only applies when: the expression involves exactly two terms Both terms are perfect squares and One square is subtracted from the other The factorisation pattern looks like this: a2 – b2 = (a – b)(a + b) Press the space bar to see examples What can I expect in the Unit test?
Unit Assessment level D.O.T.S. questions The assessment will only test to see if you have the basic idea. Take care to write out your solutions clearly and don’t take shortcuts. Press the space bar for a selection of Unit test level questions. Write out your solutions before checking the answers. Remember that your revision needs to be active. You will learn better when you do the questions not just read them. TIP! Always check for a common factor Factorise these expressions:
Factorising Trinomials 1 (unitary) A trinomial is an expression with three terms. Some typical examples are x2 + 3x – 2 or x3 – 4x2 + 3x You will be expected to rewrite trinomials in a double bracket factorised form You have to be familiar with positive and negative arithmetic You must understand how to multiply out double brackets. If you are asked to factorise an expression like x2 + 5x + 6 begin by setting out the question and two empty brackets like this: (use the space bar to go through the steps) TIP! Always check by multiplying your answer out Finally we need a factor pair for 6 that has a sum of 5 Both of the remaining terms are positive so we need + signs The first terms must be x because we have an x2 term The 3rd term is negative so the signs must be different The first terms must be x because we have an x2 term Now find a factor pair that adds up to 1 (the number of x) What can I expect in the unit test? Move on to try factorising more difficult trinomials
Assessment Style Trinomials questions The assessment will only test to see if you have the basic idea. Take care to write out your solutions clearly and always check by multiplying your solution out. Press the space bar for a selection of Unit test level questions. Write out your solutions before checking the answers. Remember that you will understand the methods when you do the questions not just read them.
Factorising Trinomials 2 (non-unitary) This section looks at how to factorise trinomials where the leading square term has a multiplying factor such as in 2x2 + 5x + 3 or 3p2 – p – 2 This isn’t assessed in the NAB but you could be asked to factorise this type in the exam. This is only one of several methods for factorising this type of trinomial. If you don’t like this method, check your textbook or online for other techniques. Suppose you have to factorise 2x2 + x – 6. The leading terms must be 2x and x and the product of the numerical terms must be -6. The factor pairs are easy to find , but the factorisation is harder because the order of the factors has to be taken into account. The only possible factor pair for 2x2 is 2x x so start with that. Next, find the factor pairs for 6 namely 16 and 23. Now use these to set up ‘cross multiplication grids’ where the rows are made from the factor pairs in all the possible combinations: For each grid form 2 new numbers by cross-multiplying the numbers on each diagonal: 2 1 6 From the x terms The factor pairs for 6 in all possible orders 2 1 6 2 1 3 2 1 3 12,1 2,6 6, 2 4,3 The pairs of numbers formed Now you have to choose the right factor pair. Since the last term is negative, the middle term is formed by subtracting these numbers. The only pair which can be subtracted to make 1 is the last one so the factor pair you need is 32 in that order. Since you need to end up with +x the largest number (4) must be positive meaning that the signs must be – then +. I’m ready to try some of these!
More Difficult Trinomial Factorisations
Completing the square 1 (unitary) This technique is very useful in the study of quadratic equations and parabolas (which comes in unit 2). In Unit 1 you just need to focus on the technique. Completing the square means rearranging a quadratic expression so that the x terms are part of a squared bracket like (x+1)2 or (x-3)2. Here’s an example of how it works: Divide the t-coefficient by 2: this gives -82 = -4. Take the -4 and make the square term (t – 4)2. Now multiply out (t – 4)2 to get t2 - 8x + 16. The 16 is not part of the given quadratic. Make sure both sides are equal by subtracting the 16 (Don’t forget the +2). Tidy up Divide the y-coefficient by 2: this gives -202 = -10. Take the -10 and make the square term (y – 10)2 . Now multiply out (y – 10)2 to get y2 – 20x + 100. The 100 is not part of the given quadratic. Make sure both sides are equal by subtracting the 100 Divide the x-coefficient by 2: this gives 82 = 4. Take the 4 and make the square term (x+4)2 . This is the ‘completing the square’ part. Now multiply out (x + 4)2 to get x2 + 8x + 16. The 16 is not part of the given quadratic. Make sure both sides are equal by subtracting the 16 What can I expect in the NAB? Harder Examples
Completing the square 2 (non-unitary) When the square term has a multiplier you need to do a couple of extra steps Factor out the x2 coefficient 2 then Complete the square on the unitary expression inside the bracket. Multiply out the original factor. Factor out the x2 coefficient 3 for the whole expression then Complete the square on the unitary expression inside the bracket. Multiply out the original factor and tidy up.
Assessment style completing the square Express the following in the form (x + a)2 + b This is just a different way of asking you to complete the square.
Mixed Course Level questions Here is a selection of questions that test your skill at factorisation and completing the square Fully factorise each of the following: Express the following in the form a(x + b)2 + c
Course Level problem solving question A square painting of side 6 cm is placed in a square wooden Frame of side k cm. Let A represent the area of the wooden frame. Write down an expression, in factorised form, for the area A cm2. What is the length of the frame if the area is 28 cm2? 6 cm k cm A = k2 – 62 = (k – 6)(k + 6) ii) A = 28 means that k2 – 36 = 28 and k2 = 64 The context requires that k is positive so the length of the frame must be 8 cm. k = 8 cm