The document provides instructions for expanding and factorizing algebraic expressions involving single and double brackets. It explains how to expand brackets by distributing terms inside brackets to each term outside. For factorizing, it describes finding common factors and grouping terms. It also covers techniques for factorizing quadratic expressions, difference of squares, and grouping. Further sections cover simplifying algebraic fractions through factorizing numerators and denominators and combining like terms.

2. Expanding single brackets Remember to multiply all the terms
inside the bracket by the term
x immediately in front of the bracket
4(2a + 3) = 8a + 12 If there is no term in
front of the bracket,
x multiply by 1 or -1
Expand these brackets and simplify wherever possible:
1. 3(a - 4) = 7. 4r(2r + 3) =
2. 6(2c + 5) = 8. - (4a + 2) =
3. -2(d + g) = 9. 8 - 2(t + 5) =
4. c(d + 4) =
10. 2(2a + 4) + 4(3a + 6) =
5. -5(2a - 3) =
11. 2p(3p + 2) - 5(2p - 1) =
6. a(a - 6) =

3. Expanding double brackets
Split the double brackets into 2
single brackets and then expand
each bracket and simplify
(3a + 4)(2a – 5)
“3a lots of 2a – 5
and 4 lots of 2a – 5”
= 6a2 – 15a + 8a – 20
If a single bracket is squared
(a + 5)2 change it into double
= 6a2 – 7a – 20 brackets (a + 5)(a + 5)
Expand these brackets and simplify :
1. (c + 2)(c + 6) = 4. (p + 2)(7p – 3) =
2. (2a + 1)(3a – 4) = 5. (c + 7)2 =
3. (3a – 4)(5a + 7) = 6. (4g – 1)2 =

4. Factorising – common factors
Factorising is basically the
Factorising
reverse of expanding brackets.
Instead of removing brackets
5x2 + 10xy = 5x(x + 2y) you are putting them in and
placing all the common factors
in front.
Expanding
Factorise the following (and check by expanding):
 15 – 3x =  10pq + 2p =
 2a + 10 =  20xy – 16x =
 ab – 5a =  24ab + 16a2 =
 a2 + 6a =  r2 + 2 r =
 8x2 – 4x =  3a2 – 9a3 =

5. Factorising – quadratic expressions
a 2  2ab  b 2  a 2  b2 
a 2  2ab  b 2  ax  by  cx  dy 

6. Factorising – grouping and difference of two squares
Grouping into pairs Difference of two squares
Fully factorise this expression: Fully factorise this expression:
6ab + 9ad – 2bc – 3cd 4x2 – 25
Factorise in 2 parts Look for 2 square numbers
separated by a minus. Simply
Use the square root of each
Rewrite as double brackets and a “+” and a “–” to get:
Fully factorise these: Fully factorise these:
(a) wx + xz + wy + yz (a) 81x2 – 1
(b) 2wx – 2xz – wy + yz (b) 4 – t2
(c) 8fh – 20fm + 6gh – 15gm (c) 16y2 - 64
Answers: Answers:
(a) (x + y)(w + z) (a) (9x + 1)(9x – 1)
(b) (2x – y)(w – z) (b) (2 + t)(2 – t)
(c) (4f + 3g)(2h – 5m) (c) 16(y2 + 4)

7. Factorise each of the following
x  6m  9
2
3r  6rp  3 p
2 2
25 x  120 xy  100 y
2 2

8. Factorise each of the following
( x  5)  9
2
4(m  1) 2  25

9. Simplifying Algebraic Fractions
Reduce this fraction
12 43 Factorise the numerator and
 denominator, cancel the
53
common factors
15
Simplify by factorising
1 Cancel the common factors
6c 2
32 c c

2c 2c
Factorise
first before
cancelling

10. Simplify by factorising
5 6 p 4m6
 
15m 5
15 p m

11. Simplify
Cancel the common factors Write down what’s left
2
7(c  1) 7(c  1)

(c  1)2 (c  1)(c  1)
Let’s do one that isn’t already factorised
3 Cancel the common factors write down what’s left
2 x  10 2( x  5)
 Grade A
3 x  15 3( x  5)
factorise the numerator first
factorise the denominator

12. Factorise and Simplify
Cancel the common factors Write down what’s left
4
( x2  9)  ( x  3)( x  3) Grade A*
x2  7 x  12 ( x  3)( x  4)
Factorise the numerator first Factorise the denominator
Simplify the expressions fully
1) 2x  6 4) x2  16
2x x2  4 x
2) 5 x  10 5) 2 x2  8
3x  6 x2  6 x  8
x2  2 x x2  5 x  6
3)
6)
8 x  16 x2  x  6

13. Check your answers!!
1) 2 x  6  2( x  3) 
x3
x Factorise the numerator
2x 2x
5 x  10  5( x  2)
2) 5 Factorise the denominator

3x  6 3( x  2) 3 Cancel the common factors
Write down what’s left
x2  2 x x( x  2) x
3)  
8 x  16 8( x  2) 2
4) x2  16 ( x  4)( x  4)
  x4
x  4x
2 x ( x  4)
5) 2 x2  8 2( x 2  4 ) 2( x  2)( x  2) 2( x  2)
  
x2  6 x  8 ( x  4)( x  2) ( x  4)( x  2) x4
x2  5 x  6 ( x  3)( x  2) 
x2
6)  x2
x2  x  6 ( x  3)( x  2)

14. Factorise the numerator first
Factorise and Simplify
Factorise the denominator
k 2  36 (a  b) 2  9b 2 Cancel the common factors
(k  6) 2 a 2  2ab Write down what’s left

15. Algebraic fractions – Addition and subtraction
Like ordinary fractions you can only add or subtract algebraic
fractions if their denominators are the same
Simplify 3 + 4_
x y
Addition
a c ad  bc
 
b d bd
Multiply the top
and bottom of
each fraction by
the same amount

16. Test yourself!!
Simplify.
x 3x
1)  =
10 10
5 2
2)  =
7x 7x
4m 2m  1
3)  =
5 5

17. Simplify x – x-5
2 6

18. Simplify
Addition
1 3 a c ad  bc
  
x 2 x b d bd

19. Simplify
Addition
3 2 a c ad  bc
  
x y yx b d bd

20. Simplify
Subtraction
2 3p a c ad  bc
2
 2  
p qr pr b d bd

21. Simplify
Subtraction
2 x 3x  3
 a c ad  bc
2 4  
b d bd

22. Algebraic fractions – Multiplication and division
Again just use normal fractions principles
Simplify:
x x 5y 2
 5t  2
y 10 y 4 x
a c ac
 
b d bd
Multiplication

23. Simplify
Multiplication
2x  y 5x
 a c ac
y 3x  y  
b d bd

24. Test yourself!!!
2x 9 y 5a 2b 3c
 2

3 10 x 6c 10 ab
3y a
2
5 4b c

25. Test yourself!!!
x  4 6x
2 2 9x  4 y
2 2
3y
 
3x x2 6y 3
6x  4 y

26. Simplify
Division
2x 4 y
 a c ad
7 14  
b d bc

27. Simplify
x y
2 2
4( x  y )
4

3y 9x3 y 2

28. Simplify
x2  y2 2x  2 y
3

3 xy x2 y