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Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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TERMINOLOGY 2 Algebra and Surds Binomial: A mathematical expression consisting of two terms such as 3 x + or x 3 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. x 2 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 , 3 x - Trinomial: An expression with three terms such as x x 3 2 1 2 - +
Transcript
Page 1: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

- +

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Page 2: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

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Page 3: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

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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

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Page 5: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

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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

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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

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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?

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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

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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

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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

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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

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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.

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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

+ + + = + + +

= + +

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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

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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.

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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+

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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

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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

Page 20: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

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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

Page 22: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

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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

Page 24: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

Page 25: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

Page 26: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

Page 27: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

Page 28: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

Page 29: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

Page 30: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

Page 31: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

Page 32: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.+

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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

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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

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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

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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

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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

+ =

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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

Page 50: Math In Focus Year 11 2 unit Ch2 Algebra and Surds

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


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