Subtle Factorising.pdf

Patrons are reminded that when they see 9x(x −1)2 −6x2(x −1) they should blur your eyes somewhat to see 9xz2 −6x2z. This clearly factories to 3xz(3z −2x) so the original factorises thus:
9x(x −1)2 −6x2(x −1) = 3x(x −1)[3(x −1) −2x] = 3x(x −1)(x −3).
Now factorise the following expressions fully.
1. x(x + 4) −3(x + 4) = (x + 4)(x −3)
2. 2x(x −3) + (x −3) = (x −3)(2x + 1)
3. 4(2x −1) + 3x(2x −1) = (2x −1)(3x + 4)
4. x(x + 2) + 2(x + 2) = (x + 2)2
5. 3x(y −1) + 2(y −1) = (y −1)(3x + 2)
6. p(x + y) + q(x + y) = (x + y)(p + q)
7. 4q(2x −y) −3p(2x −y) = (2x −y)(4q −3p)
8. 4x2(x + 2)2 −2x(x + 2) = 2x(x + 2)(2x2 + 4x −1)
9. 10x5(2x + 1) + 5x4(2x + 1)2 = 5x4(2x + 1)(4x + 1)
10. 49y(y −4)10 −35y2(y −4)9 = 14y(y −4)9(y −14)
11. (x + 1)4(x + 2) −2(x + 1)3(x + 2)2 = −(x + 2)(x + 1)3(x + 3)
12. 20(2x + 1)4(x −2)5 −25(2x + 1)5(x −2)4 = −5(6x + 13)(2x + 1)4(x −2)4
13. 16x2(x −1) −20x(x −1)2 = −4x(x −1)2
For the next few patrons are reminded that 3x(x + 1) / (x + 1)2 = 3x / (x + 1) and that factorising the top and bottom of a fraction is a wise way to proceed. Simplify the following:
14. (4x + 8) / (4) / (x + 2)
15. (7) / (14x −21) / (1) / (2x−3)
16. (3x + 6) / (5x + 10) / (3) / (5)
17. (x2 −4x) / (2x −8) / (x) / (2)
18. (4x + 2) / (6x2 + 3x) / (2) / (3x)
19. 2x(x + 1)2 / (x3 + x2) / (2(x+1)) / (x)
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