J17

(e) (3x + 2)(3x – 2) = 3x(3x – 2) + 2(3x – 2)
= 9x 2 – 6x + 6x – 4
= 9x 2 – 4
Note: 9x 2 – 4 = (3x) 2 – 2 2 .
Exercise : Expand.
1. (x + 1) 2 2. (x + 2) 2
3. (a – 5) 2 4. (p + q) 2
5. (a + 6) 2 6. (b – 8) 2
7. (x – y) 2 8. (x + 1)(x – 1)
9. (y – 9)(y + 9) 10. (y – x)(y + x)
11. (y – a)(y + a) 12. (3x + 4) 2
13. (2a – 7)(2a + 7) 14. (5x + 3) 2
15. (4x – 1) 2 16. (7a – 3) 2
17. (2x – 1) 2 18. (3 – 2x) 2
19. (c – ax) 2 20. (x – 1) 2
21. u 1 2
22. a 3
2
4
23. 2x 1 2
24. 1 x 3 2
3
2
4
Factorizing expressions
If the product of 5 and 7 is 35, then 5 and 7 are factors of 35.
In algebra, letters represent numbers. Therefore, we can extend the idea of factors to
algebraic expressions. For example, a and b are factors of ab and 2, x and y are factors
of 2xy. Also, given that 5(a – b) = 5a – 5b, then, 5 and (a – b) are factors of 5a – 5b.
In order to find the factors of an expression such as 10x 2 + 15x, we look for the factors
that are common in both terms. The common factors of 10x 2 and 15x are 5 and x.
Thus, 10x 2 + 15x = (5x × 2x) + (5x × 3)
= 5x(2x + 3).
Therefore, the factors of 10x 2 + 15x are 5, x and (2x + 3). This means,
10x 2 + 15x can be factorized as 5x(2x + 3).
The process of finding the factors of an expression is called factorization. This is the
reverse of expansion.
Consider (x + 2)(x + 3) = x(x + 3) + 2(x +3)
= x 2 + 3x + 2x + 6
= x 2 + 5x + 6.
The expressions (x + 2) and (x + 3) are the factors of x 2 + 5x + 6.