1. Creating brackets

2. In this powerpoint, we meet 5 different methods of factorising. Type 1 – Common Factor Type 2 – Difference of Two Squares Type 3 – Grouping This involves taking a term outside the brackets. Always try to do this first. Try this when you have two terms with a minus between This is the easiest one to pick – use it when there are 4 terms!

3. Types 4 and 5 Quadratic trinomials Use these for expressions with 3 terms. They will be of the format x 2 + bx + c (Type 4) OR ax 2 + bx + c (Type 5) Where a, b and c are just numbers Factorising just makes me sooooo happy!!

4. Summary Type When to Use 1. Common factor Always try first before any other method Always try first before any other method Examples: a 2 – 9a ; 2xy + 5x 2 Examples: a 2 – 9a ; 2xy + 5x 2 2. Difference of Two squares When there are only 2 terms which are squares When there are only 2 terms which are squares There must be a minus sign There must be a minus sign Examples: a 2 – 25 ; 81 – 4b 2 ; w 4 – 16 Examples: a 2 – 25 ; 81 – 4b 2 ; w 4 – 16 3. Grouping There are 4 terms. There are 4 terms. Example: a 2 – 4a + 3ab – 12b Example: a 2 – 4a + 3ab – 12b 4. Quadratic Trinomial (I) There are 3 terms. Has a squared term. There are 3 terms. Has a squared term. Examples: a 2 – 9a + 20 ; 6 – 5b + b 2 Examples: a 2 – 9a + 20 ; 6 – 5b + b 2 5. Quadratic Trinomial (II) There are 3 terms. Has a squared term with a number attached in front. There are 3 terms. Has a squared term with a number attached in front. Examples: 2a 2 – 3a – 5 ; 6b – 5b 2 + 3b Examples: 2a 2 – 3a – 5 ; 6b – 5b 2 + 3b

5. Type 1 of 5 – common factor Always try this first, regardless of what type it is 3a – 12 =3(a – 4) 3a 2 – 12a = 3a 2 + 6a + 12 = 20ab – 12b 2 = 30a 6 – 15a 5 = 3a(a – 4) 4b(5a – 3b) 15a 5 (2a – 1) 3(a 2 + 2a + 4) Remember – take out the largest factor you can! Always look for a common factor!

6. Type 2 of 5 – diff of 2 squares To qualify as a Type 2, an expression must have only 2 terms which are SQUARES must have a MINUS sign separating them Examples a 2 – 9 =(a – 3)(a + 3) 16 – a 2 =(4 – a)(4 + a) (2b) 2 – (3a) 2 = 9b 2 – 25 =(3b – 5)(3b + 5) (2b – 3a)(2b + 3a)

7. Combining Types 1 and 2 Example 1.....Factorise 5x 2 – 45 STEP 1Treat as a Type 1, and take out common factor first, 5 Write 5(x 2 – 9) STEP 2Now do expression in brackets as a Type 2 Write 5(x – 3)(x + 3)...ANS! LookMum ! It’s a difference of 2 squares!

8. Example 2.....Factorise x 4 – 81 STEP 1Treat as a Type 2, and write as difference of 2 squares..... (x 2 – 9)(x 2 + 9) STEP 2 (x 2 – 9)(x 2 + 9) (x – 3)(x + 3)(x 2 + 9)....ANS!! Now check out the thing in each bracket. We can factorise the first one, but not the second. Y’can’t factorise a SUM of two squares Stupid! x 2 + 9 has to stay as it is. It’s not the same as (x + 3)(x + 3) is it now???

9. Example 3.....Factorise 80a 4 – 405b 12 STEP 2 STEP 3 STEP 1Identify common factor, 5 and remove Write 5(16a 4 – 81b 12 ) Now work on the terms in the brackets This is a difference of 2 squares and becomes (4a 2 – 9b 6 ) (4a 2 + 9b 6 ) Now work on the terms in the 1 st bracket. This is a difference of 2 squares and becomes (2a – 3b 3 ) (2a + 3b 3 ). Write Write 5(4a 2 – 9b 6 ) (4a 2 + 9b 6 ) 5(2a – 3b 3 ) (2a + 3b 3 ) (4a 2 + 9b 6 )

10. Example 4.....Factorise 9a 2 – (x – 2a) 2 Just treat as difference of 2 squares of the format 9a 2 – b 2 where the b = [x – 2a] Factorising it then becomes = (3a – b)(3a + b) And then replacing the b with [x – 2a] we get = (3a – [x – 2a])(3a + [x – 2a]) Now get rid of square brackets = (3a – x + 2a)(3a + x – 2a) Clean up = (5a – x )(a + x) Ans!! You could check your answer by expanding it and also expanding the original question. They should both give the same thing.

11. Type 3 of 5 – Grouping You can tell when you’ve got one of these because there are FOUR TERMS !!! Example 1 Factorise 2a – 4b + ax – 2bx STEP 1 – split it into “2 by 2” = 2a – 4b + ax – 2bx STEP 2 – factorise each pair separately as Type 1 = 2(a – 2b) + x(a – 2b) STEP 3 – take out the (a – 2b) as a common factor = (a – 2b)(2 + x)...ans!! No need to be confused!

12. Type 3 of 5 – Grouping Example 2 Factorise xy + 5x – 2y – 10 STEP 1 – split it into “2 by 2” = xy + 5x – 2y – 10 STEP 2 – factorise each pair separately as Type 1 = x(y + 5) – 2 (y + 5) STEP 3 – take out the (y + 5) as a factor = (y + 5)(x – 2) ans!! If these are the same, it’s a good sign!

13. Type 3 of 5 – Grouping Example 3 Factorise x 2 – x – 5x + 5 STEP 1 – split it into “2 by 2” = x 2 – x – 5x + 5 STEP 2 – factorise each pair separately as Type 1 = x(x – 1) – 5 (x – 1) STEP 3 – take out the (x – 1) as a factor = (x – 1 )(x – 5) ans!! Ewbewdy!! They’re the same!  On my way to a VHA   

14. Example 4 - harder Factorise x 2 – 4y 2 – 2ax – 4ay STEP 1 – split it into “2 by 2” = x 2 – 4y 2 – 2ax – 4ay STEP 2 – factorise each pair separately = (x – 2y) (x + 2y) STEP 3 – take out the (x + 2y) as a factor = (x + 2y)(x – 2y – 2a) ans!! – 2a (x + 2y) 1 st pair – Type 22 nd pair – Type 1 Awwright! They’re the same!!

15. Type 4 of 5 – Easy Quadratic Trinomial Example 1.....Factorise x 2 + 5x + 6 You can usually pick these as they have 3 TERMS STEP 1 – Make 2 brackets (x..............)(x.............) STEP 2 – Look for 2 numbers that Multiply to make +6 Add to make +5 +2 & +3 STEP 3 – Put ‘em in the brackets (x + 2)(x + 3)

16. Type 4 of 5 – Easy Quadratic Trinomial Example 2.....Factorise 2x 2 – 6x – 20 STEP 1 – take out a common factor (remember this should be your 1 st step EVERY time!!) = 2(x 2 – 3x – 10) STEP 2 – Ignore the 2. For the expression inside the brackets, look for 2 numbers that Multiply to make – 10 Add to make – 3 +2 & – 5 STEP 3 – Put ‘em in the brackets 2(x + 2)(x – 5)

17. Type 4 of 5 – Easy Quadratic Trinomial Example 3.....Factorise 6 + 5x – x 2 STEP 1 – Rearrange into “normal” format with x 2 at the front, then x, then the number = – x 2 + 5x + 6 STEP 2 – Now take out a common factor – 1 STEP 3 – Ignore the minus. Look for 2 numbers that add to – 5, and multiply to – 6. = – (x 2 – 5x – 6) These are +1 and –6. – (x + 1)(x – 6)

18. Type 5 of 5 – Harder Quadratic Trinomial Example 1.....Factorise 2x 2 + 5x – 3 STEP 1 – Draw up a fraction like this STEP 2 – Look for two numbers that ADD to make +5 MULT to make – 6 2 × – 3 = – 6 Numbers are +6, – 1 = (x + 3)(2x – 1) ANS Note the 2 in bottom must cancel one whole bracket FULLY! So (2x + 6) becomes (x + 3) With a number in front of the x 2

19. Type 5 of 5 – Harder Quadratic Trinomial Example 2.....Factorise 3x 2 + 8x – 3 STEP 1 – Draw up a fraction like this STEP 2 – Look for two numbers that ADD to make +8 MULT to make – 9 3 × – 3 = – 9 Numbers are +9, – 1 = (x + 3)(3x – 1) ANS Note the 3 in bottom must cancel one whole bracket FULLY! So (3x + 9) becomes (x + 3) With a number in front of the x 2

20. Type 5 of 5 – Harder Quadratic Trinomial Example 3.....Factorise 6x 2 – 19x + 10 STEP 1 – Draw up a fraction like this STEP 2 – Look for two numbers that ADD to make –19 MULT to make 60 6 × 10 = 60 Numbers are –4, –15 = (3x – 2)(2x – 5) ANS Note the 6 in bottom would not cancel either bracket FULLY! So we broke the 6 into 2 x 3 then cancelled. With a number in front of the x 2

21. Now wozn’t that just a barrel of fun??