factorisation by grouping

1. 3NA 2.1 (pg 44)
Factorisation by Grouping

2. Recall –
1. Simplification of Linear Expressions
4𝑥
3
+
5 2𝑥 − 7
2
=
8𝑥
6
+
15 2𝑥 − 7
6
=
8𝑥 + 30𝑥 − 105
6
=
38𝑥 − 105
6
𝟐 ×
𝟐 ×
× 𝟑
× 𝟑
Expand by Rainbow Method
Merge to 1 denominator
Group together like terms
and simplify
Page 44

3. Recall –
2. Special Products
• 𝑎 + 𝑏 2
= 𝑎 + 𝑏 𝑎 + 𝑏
= 𝑎2 + 𝑎𝑏 + 𝑏𝑎 + 𝑏2
Note : 𝒃𝒂 = 𝒂𝒃
= 𝑎2 + 2𝑎𝑏 + 𝑏2
• 𝑎 − 𝑏 2
= 𝑎 − 𝑏 𝑎 − 𝑏
= 𝑎2 − 𝑎𝑏 − 𝑏𝑎 + 𝑏2
= 𝑎2 −2𝑎𝑏 + 𝑏2
• 𝑎 + 𝑏 𝑎 − 𝑏
= 𝑎2 − 𝑎𝑏 + 𝑏𝑎 − 𝑏2
= 𝑎2 − 𝑏2
Page 44

4. Recall –
2. Special Products – Examples
𝑎 𝟓𝒙 + 𝟏 2
= 𝟓𝒙 2 + 2 𝟓𝒙 𝟏 + 𝟏 2
= 25𝑥2 + 10𝑥 + 1
𝑏 𝟐𝒙 − 𝟕𝒚 2
= 𝟐𝒙 2
− 2 𝟐𝒙 𝟕𝒚 + 𝟕𝒚 2
= 4𝑥2
− 28𝑥𝑦 + 49𝑦2
𝑐 (𝟒𝒑 + 𝟗𝒒)(𝟒𝒑 − 𝟗𝒒)
= 𝟒𝒑 2
− 𝟗𝒒 2
= 16𝑝2 − 81𝑞2
Page 44

5. Recall –
3. Expansion
(3𝑥 − 1)(8𝑥 − 5)
Rainbow Method
= 24𝑥2
− 15𝑥 − 8𝑥 + 5
Combine like terms
= 24𝑥2
− 23𝑥 + 5
Page 44

6. Recall –
4. Factorisation
(a) Factorisation by Extracting Common Factors
4𝑎𝑥 + 6𝑎𝑦
Take out common factor “2a”
= 2𝑎(2𝑥 + 3𝑦)
(b) Factorisation by Cross Method
6𝑥2
+ 7𝑥 − 5
= (2𝑥 − 1)(3𝑥 + 5)
Page 44
𝟔𝒙 𝟐 −𝟓 𝟕𝒙
−𝟏
𝟓 𝟏𝟎𝒙
−𝟑𝒙𝟐𝒙
𝟑𝒙

7. Example 1
Factorise 12𝑎𝑥 + 3𝑎𝑦 + 8𝑏𝑥 + 2𝑏𝑦.
12𝑎𝑥 + 3𝑎𝑦 + 8𝑏𝑥 + 2𝑏𝑦
Arrange into 2 groups
= (12𝑎𝑥 + 3𝑎𝑦) + (8𝑏𝑥 + 2𝑏𝑦)
Take out common factor from each group
= 𝟑𝒂 𝟒𝒙 + 𝒚 + 𝟐𝒃 𝟒𝒙 + 𝒚
Take out common factor
= (𝟒𝒙 + 𝒚)(𝟑𝒂 + 𝟐𝒃)
Try it 1: Factorise 6𝑎𝑥 − 10𝑎𝑦 + 3𝑏𝑥 − 5𝑏𝑦
Answer: (3𝑥 − 5𝑦)(2𝑎 + 𝑏)

8. Example 2
Factorise 5𝑐𝑘 − 5𝑐 + 6 − 6𝑘.
5𝑐𝑘 − 5𝑐 + 6 − 6𝑘
Arrange into 2 groups
= (5𝑐𝑘 − 5𝑐) + (6 − 6𝑘)
Take out common factor from each group
= 5𝑐 𝑘 − 1 + 6(1 − 𝑘)
𝟏 − 𝒌 = −(𝒌 − 𝟏)
= 𝟓𝒄 𝒌 − 𝟏 − 𝟔(𝒌 − 𝟏)
Take out common factor
= (𝒌 − 𝟏)(𝟓𝒄 − 𝟔)
Try it 2: Factorise 7𝑎𝑥 − 21𝑥 + 12 − 4𝑎
Answer: (𝑎 − 3)(7𝑥 − 4)