3 na 2.1 factorisation by grouping part 1
- 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)