Case 2: Ensure the common factor (bracket) in each pair is identical by altering the signs. For example, let’s looks at how to factorise.
ak − at − 5k + 5t 1st pair 2nd pair
If +5 is used as the HCF for the 2nd pair, then (a)(k − t) + ( 5 )(− k + t)
Note that the 2nd bracket in each pair is not identical.
(k − t) is not equal to ( −k + t)
However, if − 5 is used as the HCF for the 2nd pair, then we get the same second bracket in each pair: (a)(k − t) + (− 5)(k − t)
To complete the factorisation, we take out the common factor: (k − t)(a − 5 )
Worked example 2 Factorise fully the expression x2
Solution + 3y – 3x – xy
(x − y) = ( − y + x)
Practice questions 20.6
1. Factorise the following expressions fully. (i) p ( x + a ) + q ( x + a ) (ii) m (x − y) + n ( x − y ) (iii) ax + bx + ay + by (iv) mx + nx + my + ny
2. Factorise the following expressions fully. (i) 5x + 5y + ax + ay (ii) am + an + 4m + 4n (iii) ac + bc + ad + bd (iv) pq + pr + xq + xr
3. (a) Factorise the following expressions fully. (i) xy + xz − y − z (ii) 2x − x2
+ 2y − xy (iii) ax + 4p− 4a − px 328
(b) Verify your factors for part (a). Linking Thinking 1
4. (a) Factorise the following expressions fully. (i) nq + nr + 4q + 4r (ii) 6my + 2ny + 3am + an (iii) ax + bx + a + b
(b) Verify your factors for part (a).
5. (a) Factorise the following expressions fully. (i) 3n + yz + 3y + nz (ii) an – 5a − 5b + bn (iii) a2
+ 2a − ab − 2b (b) Verify your factors for part (a).
6. Factorise the following expressions fully. (i) xy − 5y − 2x + 10 (ii) x + 3y + x2