Expanding brackets and factorising expressions.

Look at this algebraic expression: 4( a + b ) What do you think it means? Remember, in algebra we do not write the multiplication sign, ×. This expression actually means: 4 × ( a + b ) or ( a + b ) + ( a + b ) + ( a + b ) + ( a + b ) = a + b + a + b + a + b + a + b = 4 a + 4 b Brackets

Expanding brackets then simplifying Sometimes we need to multiply out brackets and then simplify. For example: 3 x + 2(5 – x ) We need to multiply the bracket by 2 and collect together like terms. 3x3x + 10 – 2 x = 3 x – 2 x + 10 = x + 10

Expanding brackets then simplifying Simplify: 4 – (5 n – 3) We need to multiply the bracket by –1 and collect together like terms. 4 – 5 n + 3 = – 5 n = 7 – 5 n

Expanding brackets then simplifying Simplify: 2(3 n – 4) + 3(3 n + 5) We need to multiply out both brackets and collect together like terms. 6n6n – n + 15 = 6 n + 9 n – = 15 n + 7

Simplify: 5(3 a + 2 b ) – 2(2 a + 5 b ) We need to multiply out both brackets and collect together like terms. 15 a + 10 b – 4 a –10 b = 15 a – 4 a + 10 b – 10 b = 11 a Expanding brackets then simplifying

A little note about Vocabulary… “Expand the brackets” means the same as… Remove the brackets… Simplify the brackets…

Some expressions can be simplified by dividing each term by a common factor and writing the expression using brackets. In the expression: 5 x + 10 the terms 5 x and 10 have a common factor, 5. We can write the 5 outside of a set of brackets 5( x + 2) We can write the 5 outside of a set of brackets and mentally divide 5 x + 10 by 5. (5 x + 10) ÷ 5 = x + 2 This is written inside the bracket. 5( x + 2)

Writing 5 x + 10 as 5( x + 2) is called factorizing the expression. Factorize 6 a a + 8 =2(3 a + 4) Factorize 12 – 9 n 12 – 9 n =3(4 – 3 n ) The highest common factor of 6 a and 8 is 2. (6 a + 8) ÷ 2 =3 a + 4 The highest common factor of 12 and 9 n is 3. (12 – 9 n ) ÷ 3 =4 – 3 n

Writing 5 x + 10 as 5( x + 2) is called factorizing the expression. 3 x + x 2 = x (3 + x ) 2 p + 6 p 2 – 4 p 3 = 2 p (1 + 3 p – 2 p 2 ) The highest common factor of 3 x and x 2 is x. (3 x + x 2 ) ÷ x = 3 + x The highest common factor of 2 p, 6 p 2 and 4 p 3 is 2p. (2 p + 6 p 2 – 4 p 3 ) ÷ 2 p = p – 2 p 2 Factorize 3 x + x 2 Factorize 2 p + 6 p 2 – 4 p 3

Questions on expanding brackets:

Answers: 1. 3x x x – 84. 6x – x x x x – 9x10. 8x – x – x x – – 4x15. 3x + 3y 16. 5x x x x x x x _ x + 9

Factorise each of the following: 1.6a – b n – 6a 4. 8n 2 – ab + 20bd6. 25kp – 5ap7. 12n 2 – 4n 8. 9m 3 – 3m bh – 16hc + 8hk10. 4m 4 + 8m 2

Answers: 1.6a – 3 = 3(2a – 1) 2. 5b + 25 = 5(b + 5) 3. 2n – 6a = 2(n – 3a) 4. 8n 2 – 4 = 4(2n 2 – 1) 5.16ab + 20bd = 4b(4a + 5d)6. 25kp – 5ap = 5p(5k – a) 7.12n 2 – 4n = 4n(3n – 1)8. 9m 3 – 3m 2 = 3m 2 (3m – 1) 9. 24bh – 16hc + 8hk = 8h(3b – 2c + k) 10. 4m 4 + 8m 2 = 4m 2 (m 2 + 2)