math in focus year 11 2 unit ch2 algebra and surds
TRANSCRIPT
TERMINOLOGY
2 Algebra and Surds
Binomial: A mathematical expression consisting of two terms such as 3x + or x3 1-
Binomial product: The product of two binomial expressions such as ( 3) (2 4)x x+ -
Expression: A mathematical statement involving numbers, pronumerals and symbols e.g. x2 3-
Factorise: The process of writing an expression as a product of its factors. It is the reverse operation of expanding brackets i.e. take out the highest common factor in an expression and place the rest in brackets e.g. 2 8 2( 4)y y= --
Pronumeral: A letter or symbol that stands for a number
Rationalising the denominator: A process for replacing a surd in the denominator by a rational number without altering its value
Surd: From ‘absurd’. The root of a number that has an irrational value e.g. 3 . It cannot be expressed as a rational number
Term: An element of an expression containing pronumerals and/or numbers separated by an operation such as , , or# '+ - e.g. 2 , 3x -
Trinomial: An expression with three terms such as x x3 2 12
- +
ch2.indd 44 7/17/09 11:54:59 AM
45Chapter 2 Algebra and Surds
DID YOU KNOW?
Box text...
INTRODUCTION
THIS CHAPTER REVIEWS ALGEBRA skills, including simplifying expressions, removing grouping symbols, factorising, completing the square and simplifying algebraic fractions . Operations with surds , including rationalising the denominator , are also studied in this chapter .
DID YOU KNOW?
One of the earliest mathematicians to use algebra was Diophantus of Alexandria . It is not known when he lived, but it is thought this may have been around 250 AD.
In Baghdad around 700–800 AD a mathematician named Mohammed Un-Musa Al-Khowarezmi wrote books on algebra and Hindu numerals. One of his books was named Al-Jabr wa’l Migabaloh , and the word algebra comes from the fi rst word in this title.
Simplifying Expressions
Addition and subtraction
EXAMPLES
Simplify
1. x x7 - Solution
7 7 1
6
x x x x
x
- = -
=
2. x x x4 3 62 2 2- + Solution
4 3 6 6
7
x x x x x
x
2 2 2 2 2
2
- + = +
=
Here x is called a pronumeral.
CONTINUED
ch2.indd 45 7/17/09 11:55:13 AM
46 Maths In Focus Mathematics Preliminary Course
3. x x x3 5 43 - - + Solution
x x x x x3 5 4 8 43 3- - + = - +
4. a b a b3 4 5- - - Solution
3 4 5 3 5 4
2 5
a b a b a a b b
a b
- - - = - - -
= - -
Only add or subtract ‘like’ terms. These have the same pronumeral (for example, 3 x and 5 x ).
1. 2 5x x+
2. 9 6a a-
3. 5 4z z-
4. 5a a+
5. b b4 -
6. r r2 5-
7. y y4 3- +
8. x x2 3- -
9. 2 2a a-
10. k k4 7- +
11. 3 4 2t t t+ +
12. w w w8 3- +
13. m m m4 3 2- -
14. 3 5x x x+ -
15. 8 7h h h- -
16. b b b7 3+ -
17. 3 5 4 9b b b b- + +
18. x x x x5 3 7- + - -
19. x y y6 5- -
20. a b b a8 4 7+ - -
21. 2 3xy y xy+ +
22. 2 5 3ab ab ab2 2 2- -
23. m m m5 122 - - +
24. 7 5 6p p p2 - + -
25. 3 7 5 4x y x y+ + -
26. 2 3 8ab b ab b+ - +
27. ab bc ab ac bc+ - - +
28. a x a x7 2 15 3 5 3- + - +
29. 3 4 2x xy x y x y xy y3 2 2 2 2 3- + - + +
30. 3 4 3 5 4 6x x x x x3 2 2- - + - -
2.1 Exercises
Simplify
ch2.indd 46 7/17/09 11:55:25 AM
47Chapter 2 Algebra and Surds
Multiplication
EXAMPLES
Simplify
1. x y x5 3 2# #- Solution
5 3 2 30
30
x y x xyx
x y2
# #- = -
= -
2. 3 4x y xy3 2 5#- - Solution
x y xy x y3 4 123 2 5 4 7#- - =
Use index laws to simplify this
question.
1. b5 2#
2. x y2 4#
3. p p5 2#
4. z w3 2#-
5. a b5 3#- -
6. x y z2 7# #
7. ab c8 6#
8. d d4 3#
9. a a a3 4# #
10. y3 3-^ h
11. 2x2 5^ h
12. ab a2 33 #
13. a b ab5 22 # -
14. pq p q7 32 2 2#
15. ab a b5 2 2#
16. h h4 23 7# -
17. k p p3 2#
18. t3 3 4-^ h
19. m m7 26 5# -
20. x x y xy2 3 42 3 2# #- -
2.2 Exercises
Simplify
ch2.indd 47 7/17/09 11:55:28 AM
48 Maths In Focus Mathematics Preliminary Course
Division
Use cancelling or index laws to simplify divisions.
EXAMPLES
Simplify
1. v y vy6 22 ' Solution
By cancelling,
v y vyvy
v y
v y
v v y
v
6 22
6
2
6
3
22
1 1
3 1 1
'
# #
# # #
=
=
=
Using index laws,
v y vy v y
v yv
6 2 3
33
2 2 1 1 1
1 0
' =
=
=
- -
2. 155
aba b
2
3
Solution
3
3
aba b a b
a b
ba
155
3
2
33 1 1 2
2 1
2
=
=
=
- -
-
1
1
1. x30 5'
2. y y2 '
3. 2
8a2
4. 8aa2
5. aa
28 2
6. x
xy
2
7. p p12 43 2'
8. 6
3ab
a b2 2
9. 1520
xyx
10. xx
39
4
7-
2.3 Exercises
Simplify
ch2.indd 48 7/17/09 11:55:29 AM
49Chapter 2 Algebra and Surds
11. ab b15 5'- -
12. 62a bab2 3
13. pqs
p4
8-
14. cd c d14 212 3 3'
15. 4
2
x y z
xy z3 2
2 3
16. pq
p q
7
423
5 4
17. a b c a b c5 209 4 2 5 3 1'
- - -
18. a b
a b
4
29 2 1
5 2 4
- -
-
^
^
h
h
19. x y z xy z5 154 7 8 2'- -
20. a b a b9 184 1 3 1 3'- -- -^ h
Removing grouping symbols
The distributive law of numbers is given by
a b c ab ac+ = +] g
EXAMPLE
( )7 9 11 7 20
140
# #+ =
=
Using the distributive law,
( )7 9 11 7 9 7 11
63 77
140
# # #+ = +
= +
=
EXAMPLES
Expand and simplify. 1. a2 3+] g Solution
2( 3) 2 2 3
2 6
a a
a
# #+ = +
= +
This rule is used in algebra to help remove grouping symbols.
CONTINUED
ch2.indd 49 7/17/09 11:55:30 AM
50 Maths In Focus Mathematics Preliminary Course
2. x2 5- -] g Solution
( ) ( )x x
x
x
2 5 1 2 5
1 2 1 5
2 5
# #
- - = - -
= - - -
= - +
3. a ab c5 4 32 + -] g Solution
( )a ab c a a ab a c
a a b a c
5 4 3 5 4 5 3 5
20 15 5
2 2 2 2
2 3 2
# # #+ - = + -
= + -
4. y5 2 3- +^ h Solution
( )y y
y
y
5 2 3 5 2 2 3
5 2 6
2 1
# #- + = - -
= - -
= - -
5. b b2 5 1- - +] ]g g Solution
( ) ( )b b b b
b b
b
2 5 1 2 2 5 1 1 1
2 10 1
11
# # # #- - + = + - - -
= - - -
= -
1. x2 4-] g
2. h3 2 3+] g
3. a5 2- -] g
4. x y2 3+^ h
5. x x 2-] g
6. a a b2 3 8-] g
7. ab a b2 +] g
8. n n5 4-] g
9. x y xy y3 22 2+_ i
10. k3 4 1+ +] g
11. t2 7 3- -] g
12. y y y4 3 8+ +^ h
2.4 Exercises
Expand and simplify
ch2.indd 50 7/17/09 11:55:31 AM
51Chapter 2 Algebra and Surds
13. b9 5 3- +] g
14. x3 2 5- -] g
15. m m5 3 2 7 2- + -] ]g g
16. h h2 4 3 2 9+ + -] ]g g
17. d d3 2 3 5 3- - -] ]g g
18. a a a a2 1 3 42+ - + -] ^g h
19. x x x3 4 5 1- - +] ]g g
20. ab a b a2 3 4 1- - -] ]g g
21. x x5 2 3- - -] g
22. y y8 4 2 1- + +^ h
23. a b a b+ --] ]g g
24. t t2 3 4 1 3- - + +] ]g g
25. a a4 3 5 7+ + --] ]g g
Binomial Products
A binomial expression consists of two numbers , for example 3.x + A set of two binomial expressions multiplied together is called a binomial
product. Example: x x3 2+ -] ]g g . Each term in the fi rst bracket is multiplied by each term in the second
bracket.
a b x y ax ay bx by+ + = + + +] ^g h
Proof
a b c d a c d b c d
ac ad bc bd+ + = + + +
= + + +
] ] ] ]g g g g
EXAMPLES
Expand and simplify 1. 3 4p q+ -^ ^h h
Solution
p q pq p q3 4 4 3 12+ - = - + -^ ^h h
2. 5a 2+] g
Solution
( 5)( 5)
5 5 25
10 25
a a a
a a a
a a
5 2
2
2
+ = + +
= + + +
= + +
] g
Can you see a quick way of doing this?
ch2.indd 51 7/31/09 3:43:28 PM
52 Maths In Focus Mathematics Preliminary Course
The rule below is not a binomial product (one expression is a trinomial), but it works the same way.
a b x y z ax ay az bx by bz+ + + = + + + + +] ^g h
EXAMPLE
Expand and simplify .x x y4 2 3 1+ - -] ^g h
Solution
( ) ( )x x y x xy x x y
x xy x y
4 2 3 1 2 3 8 12 4
2 3 7 12 4
2
2
+ - - = - - + - -
= - + - -
1. 5 2a a+ +] ]g g
2. x x3 1+ -] ]g g
3. 2 3 5y y- +^ ^h h
4. 4 2m m- -] ]g g
5. 4 3x x+ +] ]g g
6. 2 5y y+ -^ ^h h
7. 2 3 2x x- +] ]g g
8. 7 3h h- -] ]g g
9. 5 5x x+ -] ]g g
10. a a5 4 3 1- -] ]g g
11. 2 3 4 3y y+ -^ ^h h
12. 4 7x y- +] g h
13. 3 2x x2 + -^ ]h g
14. 2 2n n+ -] ]g g
15. 2 3 2 3x x+ -] ]g g
16. 4 7 4 7y y- +^ ^h h
17. 2 2a b a b+ -] ]g g
18. 3 4 3 4x y x y- +^ ^h h
19. 3 3x x+ -] ]g g
20. 6 6y y- +^ ^h h
21. a a3 1 3 1+ -] ]g g
22. 2 7 2 7z z- +] ]g g
23. 9 2 2x x y+ - +] g h
24. b a b3 2 2 1- + -] ]g g
25. 2 2 4x x x2+ - +] g h
26. 3 3 9a a a2- + +] g h
27. 9a 2+] g
28. 4k 2-] g
29. 2x 2+] g
30. 7y 2-^ h
31. 2 3x 2+] g
32. 2 1t 2-] g
2.5 Exercises
Expand and simplify
ch2.indd 52 7/31/09 3:43:29 PM
53Chapter 2 Algebra and Surds
33. 3 4a b 2+] g
34. 5x y 2-^ h
35. 2a b 2+] g
36. a b a b- +] ]g g
37. a b 2+] g
38. a b 2-] g
39. a b a ab b2 2+ - +] ^g h
40. a b a ab b2 2- + +] ^g h
Some binomial products have special results and can be simplifi ed quickly using their special properties. Binomial products involving perfect squares and the difference of two squares occur in many topics in mathematics. Their expansions are given below.
Difference of 2 squares
a b a b a b2 2+ - = -] ]g g
Proof
( ) ( )a b a b a ab ab b
a b
2 2
2 2
+ - = - + -
= -
a b a ab b22 2 2+ = + +] g
Perfect squares
Proof
( ) ( )
2
a b a b a b
a ab ab b
a ab b
2
2 2
2 2
+ = + +
= + + +
= + +
] g
2a b a ab b2 2 2- = - +] g
Proof
( ) ( )
2
a b a b a b
a ab ab b
a ab b
2
2 2
2 2
- = - -
= - - +
= - +
] g
ch2.indd 53 7/17/09 11:55:35 AM
54 Maths In Focus Mathematics Preliminary Course
EXAMPLES
Expand and simplify 1. 2 3x 2-] g
Solution
( )x x x
x x
2 3 2 2 2 3 3
4 12 9
2 2 2
2
- = - +
= - +
] ]g g
2. 3 4 3 4y y- +^ ^h h
Solution
(3 4)(3 4) 4
9 16
y y y
y
3 2 2
2
- + = -
= -
^ h
1. 4t 2+] g
2. 6z 2-] g
3. x 1 2-] g
4. 8y 2+^ h
5. 3q 2+^ h
6. 7k 2-] g
7. n 1 2+] g
8. 2 5b 2+] g
9. 3 x 2-] g
10. y3 1 2-^ h
11. x y 2+^ h
12. a b3 2-] g
13. 4 5d e 2+] g
14. 4 4t t+ -] ]g g
15. x x3 3- +] ]g g
16. p p1 1+ -^ ^h h 17. 6 6r r+ -] ]g g
18. x x10 10- +] ]g g
19. 2 3 2 3a a+ -] ]g g
20. 5 5x y x y- +^ ^h h
21. a a4 1 4 1+ -] ]g g
22. 7 3 7 3x x- +] ]g g
23. 2 2x x2 2+ -^ ^h h
24. 5x2 2+^ h
25. 3 4 3 4ab c ab c- +] ]g g
26. 2x x
2
+b l
27. 1 1a a a a- +b bl l
28. x y x y2 2+ - - -_ _i i6 6@ @
29. a b c 2+ +] g6 @
2.6 Exercises Expand and simplify
ch2.indd 54 7/17/09 11:55:36 AM
55Chapter 2 Algebra and Surds
30. x y1 2+ -] g7 A
31. a a3 32 2+ - -] ]g g
32. 16 4 4z z- - +] ]g g
33. 2 3 1 4x x 2+ + -] g
34. 2x y x y2+ - -^ ^h h
35. n n n4 3 4 3 2 52- + - +] ]g g
36. x 4 3-] g
37. x x x1 1 2
2 2
- - +b bl l
38. x y x y42 2 2 2 2+ -_ i
39. 2 5a 3+] g
40. x x x2 1 2 1 2 2- + +] ] ]g g g
Expand (x 4) (x 4) .- -2
PROBLEM
Find values of all pronumerals that make this true.
i i c c b
a b c
d e
f e b
i i i h g
#
Try c 9.=
Factorisation
Simple factors
Factors are numbers that exactly divide or go into an equal or larger number, without leaving a remainder.
EXAMPLES
The numbers 1, 2, 3, 4, 6, 8, 12 and 24 are all the factors of 24. Factors of 5 x are 1, 5, x and 5 x .
To factorise an expression, we use the distributive law.
a bax bx x ++ = ] g
ch2.indd 55 7/17/09 11:55:38 AM
56 Maths In Focus Mathematics Preliminary Course
EXAMPLES
Factorise
1. 3 12x + Solution
The highest common factor is 3. x x3 12 3 4+ = +] g
2. 2y y2 - Solution
The highest common factor is y. y y y y2 22 - = -^ h
3. 2x x3 2- Solution
x and x 2 are both common factors. We take out the highest common factor which is x 2 . x x x x2 23 2 2- = -] g
4. x xy5 3 32+ ++] ]g g Solution
The highest common factor is 3x + . x x x yy5 3 3 3 5 22+ + + ++ =] ] ] ^g g g h
5. 8 2a b ab3 2 3- Solution
There are several common factors here. The highest common factor is 2 ab 2 . 8 2 2 4a b ab ab a b3 2 3 2 2- = -^ h
Check answers by expanding brackets.
Divide each term by 3 to fi nd the terms inside the brackets.
ch2.indd 56 7/17/09 11:55:39 AM
57Chapter 2 Algebra and Surds
1. 2 6y +
2. x5 10-
3. 3 9m -
4. 8 2x +
5. y24 18-
6. 2x x2 +
7. 3m m2 -
8. 2 4y y2 +
9. 15 3a a2-
10. ab ab2 +
11. 4 2x y xy2 -
12. 3 9mn mn3 +
13. 8 2x z xz2 2-
14. 6 3 2ab a a2+ -
15. 5 2x x xy2 - +
16. 3 2q q5 2-
17. 5 15b b3 2+
18. 6 3a b a b2 3 3 2-
19. x m m5 7 5+ + +] ]g g
20. y y y2 1 1- - -^ ^h h 21. 4 7 3 7y x y+ - +^ ^h h
22. 6 2 5 2x a a- + -] ]g g
23. x t y t2 1 2 1+ - +] ]g g
24. a x b x3 2 2 3 2- + -] ]g g c x3 3 2- -] g
25. 6 9x x3 2+
26. 3 6pq q5 3-
27. 15 3a b ab4 3 +
28. 4 24x x3 2-
29. 35 25m n m n3 4 2-
30. 24 16a b ab2 5 2+
31. r rh2 22r r+
32. 3 5 3x x2- + -] ]g g
33. 4 2 4y x x2 + + +] ]g g
34. a a a1 1 2+ - +] ]g g
35. ab a a4 1 3 12 2+ - +^ ^h h
2.7 Exercises
Factorise
Grouping in pairs
If an expression has 4 terms, it may be factorised in pairs.
( ) ( )
( ) ( )
ax bx ay by x a b y a b
a b x y
+ + + = + + +
= + +
ch2.indd 57 7/17/09 11:55:40 AM
58 Maths In Focus Mathematics Preliminary Course
EXAMPLES
Factorise
1. 2 3 6x x x2 - + - Solution
2 3 6 ( 2) 3( 2)
( 2)( 3)x x x x x x
x x
2 - + - = - + -
= - +
2. 2 4 6 3x y xy- + - Solution
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
x y xy x y x
x y x
x y
x y xy x y x
x y x
x y
2 4 6 3 2 2 3 2
2 2 3 2
2 2 3
2 4 6 3 2 2 3 2
2 2 3 2
2 2 3
or
- + - = - + -
= - - -
= - -
- + - = - - - +
= - - -
= - -
1. 2 8 4x bx b+ + +
2. 3 3ay a by b- + -
3. x x x5 2 102 + + +
4. 2 3 6m m m2 - + -
5. ad ac bd bc- + -
6. 3 3x x x3 2+ + +
7. ab b a5 3 10 6- + -
8. 2 2xy x y xy2 2- + -
9. ay a y 1+ + +
10. 5 5x x x2 + - -
11. 3 3y ay a+ + +
12. 2 4 2m y my- + -
13. x xy xy y2 10 3 152 2+ - -
14. 4 4a b ab a b2 3 2+ - -
15. x x x5 3 152- - +
16. 7 4 28x x x4 3+ - -
17. 7 21 3x xy y- - +
18. 4 12 3d de e+ - -
19. x xy y3 12 4+- -
20. a ab b2 6 3+ - -
21. x x x3 6 183 2 +- -
22. pq p q q3 32+- -
2.8 Exercises
Factorise
ch2.indd 58 7/17/09 11:55:43 AM
59Chapter 2 Algebra and Surds
23. x x x3 6 5 103 2- - +
24. 4 12 3a b ac bc- + -
25. 7 4 28xy x y+ - -
26. x x x4 5 204 3- - +
27. x x x4 6 8 123 2- + -
28. 3 9 6 18a a ab b2 + + +
29. y xy x5 15 10 30+- -
30. r r r2 3 62r r+ - -
Trinomials
A trinomial is an expression with three terms, for example 4 3.x x2 - + Factorising a trinomial usually gives a binomial product.
x a b x a x bx ab2 + + ++ + =] ] ]g g g
Proof
( )
( ) ( )
( ) ( )
x a b x ab x ax bx abx x a b x a
x a x b
2 2+ + + = + + +
= + + +
= + +
EXAMPLES
Factorise
1. 5 6m m2 - + Solution
a b 5+ = - and 6ab = +
6235
+-
-
-
'
Numbers with sum 5- and product 6+ are 2- and 3.-
[ ] [ ]m m m mm m
5 6 2 32 3
2` - + = + - + -
= - -
] ]] ]
g gg g
2. 2y y2 + - Solution
1a b+ = + and 2ab = -
2211
-+
-+
'
Two numbers with sum 1+ and product 2- are 2+ and 1- . y y y y2 2 12` + - = + -^ ^h h
Guess and check by trying 2- and 3-
or 1- and .6-
Guess and check by trying 2 and 1- or
2- and 1.
ch2.indd 59 7/17/09 11:55:44 AM
60 Maths In Focus Mathematics Preliminary Course
The result x a b x a x bx ab2 + + + ++ =] ] ]g g g only works when the coeffi cient of x2 (the number in front of x2 ) is 1. When the coeffi cient of x2 is not 1, for example in the expression 5 2 4,x x2 - + we need to use a different method to factorise the trinomial.
There are different ways of factorising these trinomials. One method is the cross method . Another is called the PSF method . Or you can simply guess and check.
1. 4 3x x2 + +
2. y y7 122 + +
3. m m2 12 + +
4. t t8 162 + +
5. 6z z2 + -
6. 5 6x x2 - -
7. v v8 152 - +
8. 6 9t t2 - +
9. x x9 102 + -
10. 10 21y y2 - +
11. m m9 182 - +
12. y y9 362 + -
13. 5 24x x2 - -
14. 4 4a a2 - +
15. x x14 322 + -
16. 5 36y y2 - -
17. n n10 242 +-
18. x x10 252 +-
19. p p8 92 + -
20. k k7 102 +-
21. x x 122 + -
22. m m6 72 - -
23. 12 20q q2 + +
24. d d4 52 - -
25. l l11 182 +-
2.9 Exercises
Factorise
EXAMPLES
Factorise
1. 5 13 6y y2 - +
Solution—guess and check
For 5 y 2 , one bracket will have 5 y and the other y : .y y5^ ^h h Now look at the constant (term without y in it): .6+
ch2.indd 60 7/17/09 11:55:46 AM
61Chapter 2 Algebra and Surds
The two numbers inside the brackets must multiply to give 6.+ To get a positive answer, they must both have the same signs. But there is a negative sign in front of 13 y so the numbers cannot be both positive. They must both be negative. y y5 - -^ ^h h To get a product of 6, the numbers must be 2 and 3 or 1 and 6. Guess 2 and 3 and check:
3 5 15 2 6
5 17 6
y y y y y
y y
5 2 2
2
- = - - +
= - +
-^ ^h h
This is not correct. Notice that we are mainly interested in checking the middle two terms, .y y15 2and- - Try 2 and 3 the other way around: .y y5 3 2- -^ ^h h Checking the middle terms: y y y10 3 13- - = - This is correct, so the answer is .y y5 3 2- -^ ^h h Note: If this did not check out, do the same with 1 and 6.
Solution — cross method
Factors of 5y2 are 5 y and y. Factors of 6 are 1- and 6- or 2- and .3- Possible combinations that give a middle term of y13- are
By guessing and checking, we choose the correct combination.
y13-
y y
y y
5 2 10
3 3
#
#
- = -
- = -
y y y y5 13 6 5 3 22` - + = - -^ ^h h
Solution — PSF method
P: Product of fi rst and last terms 30y2 S: Sum or middle term y13- F: Factors of P that give S ,y y3 10- -
y
yyy
3031013
2 -
-
-
)
y y y y y
y y y
y y
5 13 6 5 3 10 65 3 2 5 3
5 3 2
2 2` - + = - - +
= - - -
= - -
^ ^^ ^
h hh h
5y
y 3-
2- 5y
y 2-
3- 5y
y 6-
1- 5y
y 1-
6-
5y
y 2-
3-
CONTINUED
ch2.indd 61 7/17/09 11:55:48 AM
62 Maths In Focus Mathematics Preliminary Course
2. 4 4 3y y2 + -
Solution—guess and check
For 4 y 2 , both brackets will have 2 y or one bracket will have 4 y and the other y . Try 2 y in each bracket: .y y2 2^ ^h h Now look at the constant: .3- The two numbers inside the brackets must multiply to give .3- To get a negative answer, they must have different signs. y y2 2 +-^ ^h h To get a product of 3, the numbers must be 1 and 3. Guess and check: y y2 3 2 1+-^ ^h h Checking the middle terms: y y y2 6 4- = - This is almost correct, as the sign is wrong but the coeffi cient is right (the number in front of y ). Swap the signs around:
4 6 2 3
4 4 3
y y y y y
y y
2 1 2 3 2
2
+ = +
= +
- - -
-
^ ^h h
This is correct, so the answer is .y y2 1 2 3- +^ ^h h
Solution — cross method
Factors of 4y2 are 4 y and y or 2 y and 2 y . Factors of 3 are 1- and 3 or 3- and 1. Trying combinations of these factors gives
2y# 3
2 1 2y y
y4
#- = -
= 6y
y y y y4 4 3 2 3 2 12` + - = + -^ ^h h
Solution — PSF method
P: Product of fi rst and last terms y12 2- S: Sum or middle term 4 y F: Factors of P that give S ,y y6 2+ -
y
yyy
12624
2-+
-
+
)
y y y y y
y y y
y y
4 4 3 4 6 2 32 2 3 1 2 3
2 3 2 1
2 2` + - = + - -
= + - +
= + -
^ ^
^ ^
h h
h h
2y
2y 1-
3
ch2.indd 62 8/1/09 6:13:20 PM
63Chapter 2 Algebra and Surds
Perfect squares
You have looked at some special binomial products, including 2a b a ab b2 2 2+ = + +] g and 2 .a b a ab b2 2 2- = - +] g
When factorising, use these results the other way around.
Factorise
1. a a2 11 52 + +
2. 5 7 2y y2 + +
3. x x3 10 72 + +
4. 3 8 4x x2 + +
5. 2 5 3b b2 - +
6. 7 9 2x x2 - +
7. 3 5 2y y2 + -
8. x x2 11 122 + +
9. p p5 13 62 + -
10. x x6 13 52 + +
11. y y2 11 62 - -
12. x x10 3 12 + -
13. 8 14 3t t2 - +
14. x x6 122 - -
15. 6 47 8y y2 + -
16. n n4 11 62 +-
17. t t8 18 52 + -
18. q q12 23 102 + +
19. r r8 22 62 + -
20. x x4 4 152 - -
21. y y6 13 22 +-
22. p p6 5 62 - -
23. x x8 31 212 + +
24. b b12 43 362 +-
25. x x6 53 92 - -
26. 9 30 25x x2 + +
27. 16 24 9y y2 + +
28. k k25 20 42 +-
29. a a36 12 12 +-
30. m m49 84 362 + +
2.10 Exercises
a ab b a b
a ab b a b
2
2
2 2 2
2 2 2
+ + = +
- + = -
]]
gg
ch2.indd 63 7/17/09 11:55:52 AM
64 Maths In Focus Mathematics Preliminary Course
EXAMPLES
Factorise
1. 8 16x x2 - + Solution
8 16 2(4) 4x x x x
x 4
2 2 2
2
- + = - +
= -] g
2. 4 20 25a a2 + + Solution
4 20 25 2(2 )(5) 5a a a a
a
2
2 5
2 2 2
2
+ + = + +
= +
]]
gg
Factorise
1. y y2 12 - +
2. 6 9x x2 + +
3. m m10 252 + +
4. 4 4t t2 - +
5. x x12 362 - +
6. x x4 12 92 + +
7. b b16 8 12 - +
8. a a9 12 42 + +
9. x x25 40 162 - +
10. y y49 14 12 + +
11. y y9 30 252 +-
12. k k16 24 92 +-
13. 25 10 1x x2 + +
14. a a81 36 42 +-
15. 49 84 36m m2 + +
16. t t412 + +
17. x x34
942 - +
18. yy
956
2512 + +
19. xx
2 122
+ +
20. kk
25 0 4222
- +
2.11 Exercises
In a perfect square, the constant term is always a square number.
ch2.indd 64 7/17/09 11:55:54 AM
65Chapter 2 Algebra and Surds
Difference of 2 squares
A special case of binomial products is a b a b a b2 2+ - = -] ]g g .
a b a ba b2 2 + -- = ] ]g g
EXAMPLES
Factorise
1. 36d2 -
Solution
d d
d d36 6
6 6
2 2 2=
= +
- -
-] ]g g
2. b9 12 -
Solution
( ) ( )
b bb b
9 1 3 13 1 3 1
2 2 2- = -
= + -
] g
3. ( ) ( )a b3 12 2+ - -
Solution
[( ) ( )] [( ) ( )]( ) ( )
( ) ( )
a b a b a ba b a b
a b a b
3 1 3 1 3 13 1 3 1
2 4
2 2+ - - = + + - + - -
= + + - + - +
= + + - +
] ]g g
Factorise
1. 4a2 -
2. 9x2 -
3. y 12 -
4. 25x2 -
5. 4 49x2 -
6. 16 9y2 -
7. z1 4 2-
8. t25 12 -
9. 9 4t2 -
10. x9 16 2-
11. 4x y2 2-
12. 36x y2 2-
2.12 Exercises
ch2.indd 65 7/31/09 3:43:29 PM
66 Maths In Focus Mathematics Preliminary Course
13. 4 9a b2 2-
14. x y1002 2-
15. 4 81a b2 2-
16. 2x y2 2+ -] g
17. a b1 22 2- - -] ]g g
18. z w12 2- +] g
19. x412 -
20. y
91
2
-
21. x y2 2 12 2+ - +] ^g h
22. x 14 -
23. 9 4x y6 2-
24. x y164 4-
25. 1a8 -
Sums and differences of 2 cubes
a b a ab ba b3 3 2 2+ - ++ = ] ^g h
a b a b a ab b3 3 2 2- = - + +] ^g h
Proof
( ) ( )a b a ab b a a b ab a b ab b
a b
2 2 3 2 2 2 2 3
3 3
+ - + = - + + - +
= +
Proof
( ) ( )a b a ab b a a b ab a b ab b
a b
2 2 3 2 2 2 2 3
3 3
- + + = + + - - -
= -
EXAMPLES
Factorise
1. 8 1x3 + Solution
( ) [ ( ) ( ) ]
( ) ( )
x x
x x x
x x x
8 1 2 1
2 1 2 2 1 1
2 1 4 2 1
3 3 3
2 2
2
+ = +
= + - +
= + - +
]]
gg
ch2.indd 66 7/17/09 11:55:58 AM
67Chapter 2 Algebra and Surds
Factorise
1. b 83 -
2. 27x3 +
3. 1t3 +
4. 64a3 -
5. 1 x3-
6. 8 27y3+
7. 8y z3 3+
8. 125x y3 3-
9. 8 27x y3 3+
10. 1a b3 3 -
11. 1000 8t3+
12. x8
273
-
13. a b
1000 13 3
+
14. x y1 3 3+ -] g
15. x y z216125 3 3 3+
16. 2 1a a3 3- - +] ]g g
17. x127
3
-
18. 3y x3 3+ +] g
19. x y1 23 3+ + -] ^g h
20. 8 3a b3 3+ -] g
2.13 Exercises
2. 27 64a b3 3- Solution
( ) [ ( ) ( ) ]
( ) ( )
a b a b
a b a a b b
a b a ab b
27 64 3 4
3 4 3 3 4 4
3 4 9 12 16
3 3 3 3
2 2
2 2
- = -
= - + +
= - + +
] ]] ]
g gg g
Mixed factors
Sometimes more than one method of factorising is needed to completely factorise an expression.
EXAMPLE
Factorise 5 45.x2 - Solution
5 45 5( 9) (using simple factors)
5( 3)( 3) (the difference of two squares)x x
x x
2 2- = -
= + -
ch2.indd 67 7/17/09 11:56:00 AM
68 Maths In Focus Mathematics Preliminary Course
Factorise
1. x2 182 -
2. p p3 3 362 - -
3. y5 53 -
4. 4 8 24a b a b ab a b3 2 2 2 2+ - -
5. a a5 10 52 - +
6. x x2 11 122- + -
7. z z z3 27 603 2+ +
8. ab a b9 4 3 3-
9. x x3 -
10. x x6 8 82 + -
11. m n mn3 15 5- - +
12. x x3 42 2- - +] ]g g
13. y y y5 5162 + +-^ ^h h
14. x x x8 84 3- + -
15. x 16 -
16. x x x3 103 2- -
17. x x x3 9 273 2- - +
18. 4x y y2 3 -
19. 24 3b3-
20. 18 33 30x x2 + -
21. 3 6 3x x2 - +
22. 2 25 50x x x3 2+ - -
23. 6 9z z z3 2+ +
24. 4 13 9x x4 2- +
25. 2 2 8 8x x y x y5 2 3 3 3+ - -
26. 4 36a a3 -
27. 40 5x x4-
28. a a13 364 2 +-
29. k k k4 40 1003 2+ +
30. x x x3 9 3 93 2+ - -
2.14 Exercises
DID YOU KNOW?
Long division can be used to fi nd factors of an expression. For example, 1x - is a factor of 4 5x x+ -3 . We can fi nd the other factor by dividing 4 5x x+ -3 by 1.x -
-
5
4
5 5
5 5
0
x x
x
x xx x
x x
x
x
2
3
2
-
+ +
+
-
-
2
3
2
1x - + 4 5x -g
So the other factor of 4 5x x+ -3 is 5x x2 + + 4 5 ( 1) ( 5)x x x x x3` + - = - + +2
ch2.indd 68 7/31/09 3:43:30 PM
69Chapter 2 Algebra and Surds
Completing the Square
Factorising a perfect square uses the results a ab b a b22 2 2!! + = ] g
EXAMPLES
1. Complete the square on .x x62 + Solution
Using 2 :a ab b2 2+ +
a x
ab x2 6
=
=
Substituting :a x=
xb x
b
2 6
3
=
=
To complete the square:
a ab b a b
x x x
x x x
2
2 3 3 3
6 9 3
2 2 2
2 2 2
2 2
+ + = +
+ + = +
+ + = +
]] ]
]
gg g
g
2. Complete the square on .n n102 - Solution
Using :a ab b22 2+-
a n
ab x2 10
=
=
Substituting :a n=
nb n
b
2 10
5
=
=
To complete the square:
a ab b a b
n n n
n n n
2
2 5 5 5
10 25 5
2 2 2
2 2 2
2 2
- + = -
- + = -
- + = -
]] ]
]
gg g
g
Notice that 3 is half of 6.
Notice that 5 is half of 10.
To complete the square on ,a pa2 + divide p by 2 and square it.
2 2
a pap
ap
22 2
+ + = +d dn n
ch2.indd 69 7/17/09 11:56:04 AM
70 Maths In Focus Mathematics Preliminary Course
EXAMPLES
1. Complete the square on .x x122 + Solution
Divide 12 by 2 and square it:
x x x x
x x
x
12212 12 6
12 36
6
22
2 2
2
2
+ + = + +
= + +
= +
c
]
m
g
2. Complete the square on .y y22 - Solution
Divide 2 by 2 and square it:
y y y y
y y
y
222 2 1
2 1
1
22
2 2
2
2
+ = +
= +
=
- -
-
-
c
^
m
h
Complete the square on
1. x x42 +
2. 6b b2 -
3. 10x x2 -
4. 8y y2 +
5. 14m m2 -
6. 18q q2 +
7. 2x x2 +
8. 16t t2 -
9. 20x x2 -
10. 44w w2 +
11. 32x x2 -
12. y y32 +
13. 7x x2 -
14. a a2 +
15. 9x x2 +
16. yy
2
52 -
17. k k2
112 -
18. 6x xy2 +
19. a ab42 -
20. p pq82 -
2.15 Exercises
ch2.indd 70 7/17/09 11:56:05 AM
71Chapter 2 Algebra and Surds
Simplify
1. a5
5 10+
2. t3
6 3-
3. y
6
8 2+
4. d4 2
8-
5. x x
x5 22
2
-
6. y y
y
8 16
42 - +
-
7. a aab a
32 4
2
2
-
-
8. s ss s
5 62
2
2
+ +
+ -
9. bb
11
2
3
-
-
10. p
p p
6 92 7 152
-
+ -
11. a a
a2 3
12
2
+ -
-
12. x
x xy
8
2 233 -
- -+] ]g g
13. x x
x x x6 9
3 9 272
3 2
+ +
+ - -
14. p
p p
8 1
2 3 23
2
+
- -
15. 2 2ay by ax bx
ay ax by bx
- - +
- + -
2.16 Exercises
Algebraic Fractions
Simplifying fractions
EXAMPLES
Simplify 1.
24 2x +
Solution
x x
x2
4 22
2 1
2 1
2+=
+
= +
] g
2. 82 3 2
xx x
3
2
-
- -
Solution
xx x
x x x
x x
x xx
82 3 2
2 2 4
2 1 2
2 42 1
3
2
2
2
-
- -=
- + +
+ -
=+ +
+
] ^] ]
g hg g
Factorise fi rst, then cancel.
ch2.indd 71 7/17/09 11:56:07 AM
72 Maths In Focus Mathematics Preliminary Course
Operations with algebraic fractions
EXAMPLES
Simplify
1. x x5
14
3--
+
Solution
x x x x
x x
x
51
43
204 1 3
204 4 5 15
2019
5--
+=
- +
=- - -
=- -
-] ]g g
2. b
a b abb
a27
2 104 12
253
2 2
'+
+
+
-
Solution
ba b ab
ba
ba b ab
ab
b b b
ab aa a
b
a b bab
272 10
4 1225
272 10
254 12
3 3 9
2 55 5
4 3
5 3 98
3
2 2
3
2
2
2
2
' #
#
+
+
+
-=
+
+
-
+
=+ - +
+
+ -
+
=- - +
] ^]
] ]]
] ^
g hg
g gg
g h
3. 5
22
1x x-
++
Solution
x x x xx x
x xx x
x xx
52
21
5 22 5
5 22 4 5
5 23 1
2 1-
++
=- +
+ -
=- +
+ + -
=- +
-
+
] ]] ]
] ]
] ]
g gg g
g g
g g
Do algebraic fractions the same way as ordinary fractions.
ch2.indd 72 7/17/09 11:56:09 AM
73Chapter 2 Algebra and Surds
1. Simplify
(a) 2 4
3x x+
(b) 5
1
3
2y y++
(c) 3
24
a a+-
(d) 6
32
2p p-+
+
(e) 2
53
1x x--
-
2. Simplify
(a) 2
36 3
2b a
b b2
#+ -
+
(b) 2 1
421
q q
p
p
q2
2 3
#+ +
-
+
+
(c) xyab
x y xyab a
53
212 62
2 2'
+
-
(d) x y
ax ay bx by
ab a b
x y2 2 2 2
3 3
#-
- + -
+
+
(e) x
x xx xx x
256 9
4 55 6
2
2
2
2
'-
- +
+ -
- +
3. Simplify
(a) 2 3x x+
(b) 1
1 2x x-
-
(c) 1 3a b
++
(d) 2
xx
x2
-+
(e) 1p q p q- ++
(f) 1
13
1x x+
+-
(g) 4
22
3x x2 -
-+
(h) 2 11
11
a a a2 + ++
+
(i) 2
23
11
5y y y+
-+
+-
(j) 16
212
7x x x2 2-
-- -
4. Simplify
(a) y
xx
y
yx x
4 123
6 24
9
272 82 2
3
2
# #- -
-
+
- -
(b) y y
a aya
ay
y y
4 45
43 15
52
2
2
2
2
' #- +
-
-
- - -
(c) x x
xx
x x3
39
2 84 16
32
2
#-
+-
+
-
+
(d) b
bb b
bb
b2 6
56 12
2
'+ + -
-+
(e) x x
x xx
xx
x x5 10
8 1510
92 10
5 62
2
2
2 2
' #+
- + -
-
+ +
5. Simplify
(a) 7 101
2 152
64
x x x x x x2 2 2- +-
- -+
+ -
(b) 4
52
32
2x x x2 -
--
-+
(c) 2 3p pq pq q2 2+
+-
(d) 1a b
aa b
ba b2 2+
--
+-
(e) x yx y
y xx
y x
y2 2-
++
--
-
2.17 Exercises
Substitution
Algebra is used in writing general formulae or rules. For example, the formula A lb= is used to fi nd the area of a rectangle with length l and breadth b . We can substitute any values for l and b to fi nd the area of different rectangles.
ch2.indd 73 7/17/09 11:56:11 AM
74 Maths In Focus Mathematics Preliminary Course
EXAMPLES
1. P l b2 2= + is the formula for fi nding the perimeter of a rectangle with length l and breadth b . Find P when .l 1 3= and . .b 3 2= Solution
. .
. .
P l b2 2
2 1 3 2 3 22 6 6 4
9
= +
= +
= +
=
] ]g g
2. V r h2r= is the formula for fi nding the volume of a cylinder with radius r and height h . Find V (correct to 1 decimal place) when 2.1r = and 8.7.h = Solution
. ( . )120.5
V r h
2 1 8 7correct to 1 decimal place
2
2
r
r
=
=
=
] g
3. If F C
59 32= + is the formula for changing degrees Celsius °C] g into
degrees Fahrenheit °F] g fi nd F when 25.C = Solution
F C5
9 32
525
32
5225 32
5225 160
5385
77
9
= +
= +
= +
=+
=
=
] g
This means that °25 C is the same as .°77 F
ch2.indd 74 7/17/09 11:56:13 AM
75Chapter 2 Algebra and Surds
1. Given 3.1a = and 2.3b = - fi nd, correct to 1 decimal place.
(a) ab 3 (b) b (c) a5 2
(d) ab3
(e) a b 2+] g
(f) a b-
(g) b2-
2. T a n d1= + -] g is the formula for fi nding the term of an arithmetic series. Find T when ,a n4 18= - = and .d 3=
3. Given ,y mx b= + the equation of a straight line, fi nd y if ,m x3 2= = - and 1.b = -
4. If 100 5h t t2= - is the height of a particle at time t , fi nd h when 5.t =
5. Given vertical velocity ,v gt= - fi nd v when 9.8g = and 20.t =
6. If 2 3y x= + is the equation of a function, fi nd y when 1.3,x = correct to 1 decimal place.
7. S r r h2r= +] g is the formula for the surface area of a cylinder. Find S when 5r = and 7,h = correct to the nearest whole number.
8. A r2r= is the area of a circle with radius r . Find A when 9.5,r = correct to 3 signifi cant fi gures.
9. Given u ar 1n
n= - is the n th term of a geometric series, fi nd un if 5,a = 2r = - and 4.n =
10. Given 3V lbh= 1 is the volume
formula for a rectangular pyramid, fi nd V if . , .l b4 7 5 1= = and 6.5.h =
11. The gradient of a straight line is
given by .m x xy y
2 1
2 1=
-
- Find m
if , ,x x y3 1 21 2 1= = - = - and 5.y2 =
12. If 2A h a b= +1 ] g gives the area
of a trapezium, fi nd A when , .h a7 2 5= = and 3.9.b =
13. Find V if 3V r3r= 4 is the volume
formula for a sphere with radius r and 7.6,r = to 1 decimal place.
14. The velocity of an object at a certain time t is given by the formula .v u at= + Find v when
4 5,u a= =1 3 and 6 .t = 5
15. Given 1
,Sr
a=
- fi nd S if 5a =
and 3 .r = 2 S is the sum to infi nity of a geometric series.
16. ,c a b2 2= + according to Pythagoras’ theorem. Find the value of c if 6a = and 8.b =
17. Given 16y x2= - is the equation of a semicircle, fi nd the exact value of y when 2.x =
2.18 Exercises
ch2.indd 75 7/17/09 11:56:22 AM
76 Maths In Focus Mathematics Preliminary Course
18. Find the value of E in the energy equation E mc2= if 8.3m = and 1.7.c =
19. 1100
A P r n
= +c m is the formula
for fi nding compound interest. Find A when ,P r200 12= = and 5,n = correct to 2 decimal places.
20. If Srra
11
=-
-n^ h is the sum of
a geometric series, fi nd S if ,a r3 2= = and 5.n =
21. Find the value of c
a b2
3 2
if
4 3,a b2 3
= =3 2c cm m and .c21 4
= c m
Surds
An irrational number is a number that cannot be written as a ratio or fraction (rational). Surds are special types of irrational numbers, such as 2, 3 and 5 .
Some surds give rational values: for example, 9 3.= Others, like 2, do not have an exact decimal value. If a question involving surds asks for an exact answer, then leave it as a surd rather than giving a decimal approximation.
Simplifying surds
a b ab
a bb
aba
#
'
=
= =
Class Investigations
Is there an exact decimal equivalent for 1. 2 ? Can you draw a line of length exactly 2. 2 ? Do these calculations give the same results? 3.
(a) 9 4# and 9 4#
(b) 9
4 and
94
(c) 9 4+ and 9 4+
(d) 9 4- and 9 4-
Here are some basic properties of surds.
x x x2 2= =^ h
ch2.indd 76 7/17/09 11:56:25 AM
77Chapter 2 Algebra and Surds
EXAMPLES
1. Express in simplest surd form 45 . Solution
45 9 5
9 5
3 5
3 5
#
#
#
=
=
=
=
2. Simplify 3 40 . Solution
3 3
3
3 2
6
40 4 10
4 10
10
10
#
# #
# #
=
=
=
=
3. Write 5 2 as a single surd. Solution
5 2 25 2
50
#=
=
54 also equals 3 15# but this will
not simplify. We look for a number that is a
perfect square.
Find a factor of 40 that is a perfect square.
1. Express these surds in simplest surd form.
(a) 12
(b) 63
(c) 24
(d) 50
(e) 72
(f) 200
(g) 48
(h) 75
(i) 32
(j) 54
(k) 112
(l) 300
(m) 128
(n) 243
(o) 245
(p) 108
(q) 99
(r) 125
2. Simplify
(a) 2 27
(b) 5 80
2.19 Exercises
ch2.indd 77 7/17/09 11:56:27 AM
78 Maths In Focus Mathematics Preliminary Course
(c) 4 98
(d) 2 28
(e) 8 20
(f) 4 56
(g) 8 405
(h) 15 8
(i) 7 40
(j) 8 45
3. Write as a single surd.
(a) 3 2
(b) 2 5
(c) 4 11
(d) 8 2
(e) 5 3
(f) 4 10
(g) 3 13
(h) 7 2
(i) 11 3
(j) 12 7
4. Evaluate x if
(a) 3 5x =
(b) 2 3 x=
(c) 3 7 x=
(d) 5 2 x=
(e) 2 11 x=
(f) 7 3x =
(g) 4 19 x=
(h) 6 23x =
(i) 5 31 x=
(j) 8 15x =
Addition and subtraction
Calculations with surds are similar to calculations in algebra. We can only add or subtract ‘like terms’ with algebraic expressions. This is the same with surds.
EXAMPLES
1. Simplify 3 2 4 2 .+ Solution
3 4 72 2 2+ =
2. Simplify 3 12 .- Solution
First, change into ‘like’ surds.
3 12 3 4 3
3 2 3
3
#- = -
= -
= -
3. Simplify 2 2 2 3 .- + Solution
2 2 2 3 2 3- + = +
ch2.indd 78 7/17/09 11:56:28 AM
79Chapter 2 Algebra and Surds
Multiplication and division
Simplify
1. 5 2 5+
2. 3 2 2 2-
3. 3 5 3+
4. 7 3 4 3-
5. 5 4 5-
6. 4 6 6-
7. 2 8 2-
8. 5 4 5 3 5+ +
9. 2 2 2 3 2- -
10. 5 45+
11. 8 2-
12. 3 48+
13. 12 27-
14. 50 32-
15. 28 63+
16. 2 8 18-
17. 3 54 2 24+
18. 90 5 40 2 10- -
19. 4 48 3 147 5 12+ +
20. 3 2 8 12+ -
21. 2863 50--
22. 12 45 48 5-- -
23. 150 45 24+ +
24. 32 243 50 147-- +
25. 80 3 245 2 50- +
2.20 Exercises
To get a b c d ac bd ,# = multiply surds with surds and
rationals with rationals.
a b ab
a b c d ac bd
a a a a2
#
#
#
=
=
= =
EXAMPLES
Simplify 1. 2 2 5 7#- Solution
2 2 5 7 10 14#- = -
b
aba
=
CONTINUED
ch2.indd 79 7/17/09 11:56:31 AM
80 Maths In Focus Mathematics Preliminary Course
2. 4 2 5 18# Solution
4 2 5 18 20 36
20 6
120
#
#
=
=
=
3. 4 2
2 14
Solution
4 2
2 14
4 2
2 2 7
27
#=
=
4. 15 2
3 10
Solution
15 2
3 10
15 2
3 5 2
55
# #=
=
5. 310 2
d n
Solution
33
310
3
10
310
2
2
2
=
=
= 1
d ^^n h
h
ch2.indd 80 7/17/09 11:56:35 AM
81Chapter 2 Algebra and Surds
Simplify
1. 7 3#
2. 3 5#
3. 2 3 3#
4. 5 7 2 2#
5. 3 3 2 2#-
6. 5 3 2 3#
7. 4 5 3 11#-
8. 2 7 7#
9. 2 3 5 12#
10. 6 2#
11. 28 6#
12. 3 2 5 14#
13. 10 2 2#
14. 2 6 7 6#-
15. 22^ h
16. 2 72^ h
17. 3 5 2# #
18. 2 3 7 5# #-
19. 2 6 3 3# #
20. 2 5 3 2 5 5# #- -
21. 2 2
4 12
22. 3 6
12 18
23. 10 2
5 8
24. 2 12
16 2
25. 5 10
10 30
26. 6 20
2 2
27. 8 10
4 2
28. 3 15
3
29. 8
2
30. 6 10
3 15
31. 5 8
5 12
32. 10 10
15 18
33. 2 6
15
34. 32 2
d n
35. 75 2
d n
2.21 Exercises
Expanding brackets
The same rules for expanding brackets and binomial products that you use in algebra also apply to surds.
ch2.indd 81 7/17/09 11:56:37 AM
82 Maths In Focus Mathematics Preliminary Course
Simplifying surds by removing grouping symbols uses these general rules.
b c ab aca + = +^ h
Proof
a b c a b a c
ab ac
# #+ = +
= +
^ h
Binomial product:
a b c d ad bdac bc+ + = + + +^ ^h h
Proof
a b c d a c a d b c b d
ac ad bc bd
# # # #+ + = + + +
= + + +
^ ^h h
Perfect squares:
a b a ab b22
+ = + +^ h
Proof
a b a b a b
a ab ab ba ab b2
2
2 2
+ = + +
= + + +
= + +
^ ^ ^h h h
a b a ab b22
- = - +^ h
Proof
a b a b a b
a ab ab ba ab b2
2
2 2
- = - -
= - - +
= - +
^ ^ ^h h h
Difference of two squares:
a b a b a b+ - = -^ ^h h
Proof
a b a b a ab ab ba b
2 2+ - = - + -
= -
^ ^h h
ch2.indd 82 7/17/09 11:56:40 AM
83Chapter 2 Algebra and Surds
EXAMPLES
Expand and simplify 1. 2 5 2+^ h Solution
( )2 5 2 2 5 2 2
10 4
10 2
# #+ = +
= +
= +
2. 3 7 2 3 3 2-^ h Solution
( )3 7 2 3 3 2 3 7 2 3 3 7 3 2
6 21 9 14
# #- = -
= -
3. 2 3 5 3 2+ -^ ^h h Solution
( ) ( )2 3 5 3 2 2 3 2 2 3 5 3 3 5 2
6 2 3 15 3 10
# # # #+ - = - + -
= - + -
4. 5 2 3 5 2 3+ -^ ^h h Solution
( 2 )( 2 ) 2 2 2 2
5 2 2 4 35 12
7
5 3 5 3 5 5 5 3 3 5 3 3
15 15
# # # #
#
+ - = - + -
= - + -
= -
= -
Another way to do this question is by using the difference of two squares.
( ) ( )5 2 3 5 2 3 5 2 3
5 4 3
7
2 2
#
+ - = -
= -
= -
^ ^h h
Notice that using the difference of two
squares gives a rational answer.
ch2.indd 83 7/17/09 11:56:43 AM
84 Maths In Focus Mathematics Preliminary Course
1. Expand and simplify (a) 5 32 +^ h
(b) 2 2 53 -^ h (c) 3 3 2 54 +^ h
(d) 5 2 2 37 -^ h
(e) 3 2 4 6-- ^ h
(f) 3 5 11 3 7+^ h
(g) 3 2 2 4 3- +^ h
(h) 5 5 35 -^ h
(i) 3 12 10+^ h
(j) 2 3 18 3+^ h
(k) 4 2 3 62- -^ h
(l) 7 3 20 2 35- - +^ h
(m) 10 3 2 2 12-^ h
(n) 5 22 +- ^ h
(o) 2 3 2 12-^ h
2. Expand and simplify (a) 2 3 5 3 3+ +^ ^h h
(b) 5 2 2 7- -^ ^h h
(c) 2 5 3 2 5 3 2+ -^ ^h h
(d) 3 10 2 5 4 2 6 6- +^ ^h h
(e) 2 5 7 2 5 3 2- -^ ^h h
(f) 5 6 2 3 5 3+ -^ ^h h
(g) 7 3 7 3+ -^ ^h h
(h) 2 3 2 3- +^ ^h h
(i) 6 3 2 6 3 2+ -^ ^h h
(j) 3 5 2 3 5 2+ -^ ^h h
(k) 8 5 8 5- +^ ^h h
(l) 2 9 3 2 9 3+ -^ ^h h
(m) 2 11 5 2 2 11 5 2+ -^ ^h h
(n) 5 22
+^ h
(o) 2 2 32
-^ h
(p) 3 2 72
+^ h
(q) 2 3 3 52
+^ h
(r) 7 2 52
-^ h
(s) 2 8 3 52
-^ h
(t) 3 5 2 22
+^ h
3. If 3 2a = , simplify (a) a 2 2 (b) a 3 (2 (c) a ) 3 (d) 1a 2+] g (e) –a a3 3+] ]g g
4. Evaluate a and b if (a) 2 5 1 a b
2+ = +^ h
(b) 2 2 5 2 3 5- -^ ^h h a b 10= +
5. Expand and simplify (a) a a3 2 3 2+ - + +^ ^h h (b) 1p p
2- -_ i
6. Evaluate k if .k2 7 3 2 7 3- + =^ ^h h
7. Simplify .x y x y2 3+ -_ _i i
8. If 2 3 5 a b2
- = -^ h , evaluate a and b.
9. Evaluate a and b if .a b7 2 3 2
2- = +^ h
10. A rectangle has sides 5 1+ and 2 5 1- . Find its exact area.
2.22 Exercises
Rationalising the denominator
Rationalising the denominator of a fractional surd means writing it with a rational number (not a surd) in the denominator. For example, after
rationalising the denominator, 5
3 becomes 5
3 5.
ch2.indd 84 7/17/09 11:56:46 AM
85Chapter 2 Algebra and Surds
Squaring a surd in the denominator will rationalise it since .x x2=^ h
DID YOU KNOW?
A major reason for rationalising the denominator used to be to make it easier to evaluate the fraction (before calculators were available). It is easier to divide by a rational number than an irrational one; for example,
5
33 2.236'=
5
3 53 2.236 5# '=
This is hard to do without a calculator.
This is easier to calculate.
b
ab
bb
a b# =
Multiplying by b
b
is the same as multiplying by 1.
Proof
ba
b
b
b
a b
ba b
2# =
=
EXAMPLES
1. Rationalise the denominator of 5
3 . Solution
5
35
55
3 5# =
2. Rationalise the denominator of 5 3
2 . Solution
5 32
3
3
5 9
2 3
5 32 3
152 3
#
#
=
=
=
Don’t multiply by
5 3
5 3 as it takes
longer to simplify.
ch2.indd 85 7/17/09 11:56:50 AM
86 Maths In Focus Mathematics Preliminary Course
When there is a binomial denominator, we use the difference of two squares to rationalise it, as the result is always a rational number.
To rationalise the denominator of c d
a b
+
+ , multiply by
c d
c d
-
-
Proof
c d
a b
c d
c d
c d
a b c d
c d
a b c d
c da b c d
c d
2 2
#+
+
-
-=
+ -
+ -
=-
+ -
=-
+ -
^ ^^ ^
^ ^^ ^
^ ^
h hh h
h hh h
h h
EXAMPLES
1. Write with a rational denominator
.2 3
5
-
Solution
2 3
5
2 3
2 3
2 3
5 2 3
2 910 3 5
710 3 5
710 3 5
2 2#
- +
+=
-
+
=-
+
=-
+
= -+
^^h
h
2. Write with a rational denominator
3 4 2
2 3 5.
+
+
Solution
3 4 2
2 3 5
3 4 2
3 4 2
3 4 2
2 3 5 3 4 2
3 16 22 3 8 6 15 4 10
2 2#
#
#
+
+
-
-=
-
+ -
=-
- + -
^ ^^ ^
h hh h
Multiply by the conjugate surd 2 3.+
ch2.indd 86 7/17/09 11:56:53 AM
87Chapter 2 Algebra and Surds
296 8 6 15 4 10
296 8 6 15 4 10
=-
- + -
=- + - +
3. Evaluate a and b if .a b3 2
3 3
-= +
Solution
3 2
3 3
3 2
3 2
3 2 3 2
3 3 3 2
3 2
3 9 3 6
3 23 3 3 6
19 3 6
9 3 6
9 9 6
9 54
2 2
#
#
#
- +
+=
- +
+
=-
+
=-
+
=+
= +
= +
= +
^ ^^
^ ^
h hh
h h
.a b9 54So and= =
4. Evaluate as a fraction with rational denominator
3 2
23 2
5.
++
-
Solution
3 22
3 22
3 2
5
3 2 3 2
3 2
3 2
2 3 4 15 2 5
3 42 3 4 15 2 5
12 3 4 15 2 5
2 3 4 15 2 5
5
2 2
++
-=
+ -
- +
=-
- + +
=-
- + +
=-
- + +
= - + - -
+
^ ^^ ^
^
h hh h
h
ch2.indd 87 7/17/09 11:56:56 AM
88 Maths In Focus Mathematics Preliminary Course
1. Express with rational denominator
(a) 7
1
(b) 2 2
3
(c) 5
2 3
(d) 5 2
6 7
(e) 3
1 2+
(f) 2
6 5-
(g) 5
5 2 2+
(h) 2 7
3 2 4-
(i) 4 5
8 3 2+
(j) 7 5
4 3 2 2-
2. Express with rational denominator
(a) 3 2
4+
(b) 2 7
3
-
(c) 5 2 6
2 3
+
(d) 3 4
3 4
+
-
(e) 3 2
2 5
-
+
(f) 2 5 3 2
3 3 2
+
+
3. Express as a single fraction with rational denominator
(a) 2 1
12 1
1+
+-
(b) 2 3
2
2 33
--
+
(c) 5 2
13 2 5
3+
+-
(d) 2 3
2 7
2 3 2
2#
+
-
+
(e) tt1
+ where t 3 2= -
(f) zz12
2- where z 1 2= +
(g) 6 3
3 2 4
6 3
2 1
6 12
-
++
+
--
-
(h) 2
2 3
31+
+
(i) 2 3
3
3
2
++
(j) 6 2
5
5 32
+-
(k) 4 3
2 7
4 3
2
+
+-
-
(l) 3 2
5 2
3 1
2 3
-
--
+
+
4. Find a and b if
(a) ba
2 53
=
(b) b
a
4 2
3 6=
(c) a b5 1
2 5+
= +
(d) a b7 4
2 77
-= +
(e) a b2 1
2 3
-
+= +
2.23 Exercises
ch2.indd 88 7/17/09 11:56:59 AM
89Chapter 2 Algebra and Surds
5. Show that 2 1
2 1
24
+
-+ is
rational.
6. If x 3 2= + , simplify
(a) 1x x+
(b) 1xx
22
+
(c) 1x x
2
+b l
7. Write 5 2
25 2
1+
+-
-
3
5 1+ as a single fraction with
rational denominator .
8. Show that 3 2 2
22
8+
+ is
rational .
9. If x2 1 3+ = , where ,x 0!
fi nd x as a surd with rational denominator .
10. Rationalise the denominator of
2b
b 2
-
+ b 4!] g
ch2.indd 89 7/17/09 11:57:03 AM
90 Maths In Focus Mathematics Preliminary Course
1. Simplify (a) y y5 7-
(b) a3
3 12+
(c) k k2 33 2#-
(d) xy
3 5+
(e) 4 3 5a b a b- - - (f) 8 32+ (g) 3 5 20 45- +
2. Factorise (a) 36x2 - (b) 2 3a a2 + - (c) 4 8ab ab2 - (d) y xy x5 15 3- + - (e) 4 2 6n p- + (f) 8 x3-
3. Expand and simplify (a) bb 23 -+ ] g (b) x x2 1 3- +] ]g g (c) m m5 3 2+ --] ]g g (d) 4 3x 2-] g (e) 5 5p p- +^ ^h h (f) a a47 2 5+- -] g (g) 2 2 53 -^ h (h) 3 7 3 2+ -^ ^h h
4. Simplify
(a) b
aa
b5
4 1227
103 3#
-
-
(b) m m
mmm
25 10
3 34
2
2
'- -
+
+
-
5. The volume of a cube is .V s3= Evaluate V when 5.4.s =
6. (a) Expand and simplify .2 5 3 2 5 3+ -^ ^h h
Rationalise the denominator of (b)
.2 5 3
3 3
+
7. Simplify .x x x x2
33
16
22-
++
-+ -
8. If ,a b4 3= = - and ,c 2= - fi nd the value of
(a) ab2 (b) a bc- (c) a (d) bc 3] g (e) c a b2 3+] g
9. Simplify
(a) 6 15
3 12
(b) 2 2
4 32
10. The formula for the distance an object falls is given by 5 .d t2= Find d when 1.5.t =
11. Rationalise the denominator of
(a) 5 3
2
(b) 2
1 3+
12. Expand and simplify (a) 3 2 4 3 2- -^ ^h h (b) 7 2
2+^ h
13. Factorise fully (a) 3 27x2 - (b) x x6 12 182 - - (c) 5 40y3 +
Test Yourself 2
ch2.indd 90 7/17/09 5:04:47 PM
91Chapter 2 Algebra and Surds
14. Simplify
(a) 9
3
xy
x y5
4
(b) 15 5
5x -
15. Simplify
(a) 3 112^ h
(b) 2 33^ h
16. Expand and simplify (a) a b a b+ -] ]g g (b) a b 2+] g (c) a b 2-] g
17. Factorise (a) 2a ab b2 2- + (b) a b3 3-
18. If 3 1,x = + simplify 1x x+ and give your answer with a rational denominator.
19. Simplify
(a) 4 3a b+
(b) 2
35
2x x--
-
20. Simplify 5 2
32 2 1
2,
+-
- writing
your answer with a rational denominator.
21. Simplify (a) 3 8 (b) 2 2 4 3#- (c) 108 48-
(d) 2 18
8 6
(e) 5 3 2a b a# #- -
(f) 62m nm n
2 5
3
(g) 3 2x y x y- - -
22. Expand and simplify (a) 2 3 22 +^ h (b) 5 7 3 5 2 2 3- -^ ^h h (c) 3 2 3 2+ -^ ^h h (d) 4 3 5 4 3 5- +^ ^h h (e) 3 7 2
2-^ h
23. Rationalise the denominator of
(a) 7
3
(b) 5 3
2
(c) 5 1
2-
(d) 3 2 3
2 2
+
(e) 4 5 3 3
5 2
-
+
24. Simplify
(a) x x53
22
--
(b) a a7
23
2 3++
-
(c) x x1
11
22 -
-+
(d) 2 34
31
k k k2 + -+
+
(e) 2 5
3
3 25
+-
-
25. Evaluate n if (a) 108 12 n- =
(b) 112 7 n+ =
(c) 2 8 200 n+ =
(d) 4 147 3 75 n+ =
(e) n2 2452180
+ =
ch2.indd 91 7/17/09 11:57:20 AM
92 Maths In Focus Mathematics Preliminary Course
26. Evaluate xx12
2+ if x
1 2 3
1 2 3=
-
+
27. Rationalise the denominator of 2 7
3
(there may be more than one answer).
(a) 2821
(b) 28
2 21
(c) 1421
(d) 721
28. Simplify .x x5
34
1--
+
(a) x20
7+- ] g
(b) 20
7x +
(c) x20
17+
(d) x20
17- +] g
29. Factorise 4 4x x x3 2- - + (there may be more than one answer).
(a) x x1 42 - -^ ]h g (b) x x1 42 + -^ ]h g (c) x x 42 -] g (d) x x x4 1 1- + -] ] ]g g g
30. Simplify .3 2 2 98+ (a) 5 2 (b) 5 10 (c) 17 2 (d) 10 2
31. Simplify .x x x4
32
22
12 -
+-
-+
(a) 2 2
5x x
x+ -
+
] ]g g
(b) 2 2
1x x
x+ -
+
] ]g g
(c) 2 2
9x x
x+ -
+
] ]g g
(d) 2 2
3x x
x+ -
-
] ]g g
32. Simplify .ab a ab a5 2 7 32 2- - - (a) 2ab a2+ (b) ab a2 5 2- - (c) a b13 3- (d) 2 5ab a2- +
33. Simplify .2780
(a) 3 3
4 5
(b) 9 3
4 5
(c) 9 3
8 5
(d) 3 3
8 5
34. Expand and simplify .x y3 2 2-^ h (a) x xy y3 12 22 2- - (b) x xy y9 12 42 2- - (c) 3 6 2x xy y2 2- + (d) x xy y9 12 42 2- +
35. Complete the square on .a a162 - (a) a a a16 16 42 2- + = -^ h (b) a a a16 64 82 2- + = -^ h (c) a a a16 8 42 2- + = -^ h (d) a a a16 4 22 2- + = -^ h
ch2.indd 92 7/17/09 11:57:32 AM
93Chapter 2 Algebra and Surds
1. Expand and simplify (a) ab a b b aa4 2 32 2- --] ]g g (b) 2 2y y2 2- +_ _i i (c) x2 5 3-] g
2. Find the value of x y+ with rational denominator if x 3 1= + and
2 5 3
1 .y =-
3. Simplify .7 6 54
2 3
-
4. Complete the square on .x ab x2 +
5. Factorise (a) ( ) ( )x x4 5 42+ + + (b) 6x x y y4 2 2- - (c) 125 343x3 + (d) 2 4 8a b a b2 2- - +
6. Complete the square on .x x4 122 +
7. Simplify .x x
xy x y
4 16 12
2 2 6 62 - +
+ - -
8. | |
da b
ax by c2 2
1 1=
+
+ + is the formula for
the perpendicular distance from a point to a line. Find the exact value of d with a rational denominator if , , ,a b c x2 1 3 41= = - = = - and 5.y1 =
9. Simplify 1
.a
a
13
3
+
+^ h
10. Factorise .x b
a42 2
2
-
11. Simplify .x
x y
x
x y
x x
x y
3
2
3 6
3 22-
++
+
--
+ -
+
12. (a) Expand .x2 1 3-^ h
Simplify (b) .x x x
x x8 12 6 1
6 5 43 2
2
- + -
+ -
13. Expand and simplify 3x x1 2- -] ^g h .
14. Simplify and express with rational
denominator .3 4
2 5
2 1
5 3
+
+-
-
15. Complete the square on 3 .x x2 + 2
16. If ,xk l
lx kx1 2=
+
+ fi nd the value of x when
, ,k l x3 2 51= = - = and 4.x2 =
17. Find the exact value with rational
denominator of x x x2 3 12 - + if 2 5 .x =
18. Find the exact value of
(a) xx12
2+ if x
1 2 3
1 2 3=
-
+
(b) a and b if a b2 3 3
3 43
+
-= +
19. 2A r2i= 1 is the area of a sector of a
circle. Find the value of i when A 12= and 4.r =
20. If V r h2r= is the volume of a cylinder, fi nd the exact value of r when 9V = and 16.h =
21. If 2 ,s u at2= + 1 fi nd the exact value of s
when ,u a2 3= = and 2 3 .t =
Challenge Exercise 2
ch2.indd 93 7/17/09 11:57:46 AM