1

Number and Algebra → PatternsandAlgebra � Factorise algebraic expressions by taking out a common algebraic factor.

y use the distributive law and the index laws to factorise algebraic expressions. y understand the relationship between factorisation and expansion.

� Simplify algebraic products and quotients using index laws. y apply knowledge of index laws to algebraic terms, and simplify algebraic

expressions using both positive and negative integral indices. � Apply the four operations to simple algebraic fractions with numerical denominators.

y express the sum and difference of algebraic fractions with a common denominator.

y use the index laws to simplify products and quotients of algebraic fractions.

Can you solve the problem? A rectangle has an area of 12 and the breadth is three-quarters the length. What is the length?

The square root of a perfect square is a whole number.

A LITTLE BIT OF HISTORYAn ancient document, called the Moscow papyrus, consists of twenty-five mathematical problems and their solutions.

The document was discovered in the Necropolis of Dra Abu'l Neggra in Egypt and is estimated to have been written around 1850 BC.

If you are told: An enclosure of a set and 2 arurae, the breadth having ¾ of the length.

A TASK x1=22 + 32 + 62

x2=32 + 42 + 122

x3=42 + 52 + 202

Show that xn is a perfect square.

9 3=64 8=

36 6=

2

A convenient way of writing 2×2×2 is

Exercise1.1Write the following in index form: 2a×2×a×2a×a×a = 23a5

cddcdcccd = c5d4

1 2b×2×2b 2 abbaaabb 3 3×3x×3x×3×3x×34 xyyyxxxyy 5 10d×10d×10d×10d 6 5pp55p5ppp5

Simplify and write the following in index form:103×102 = 10×10×10 × 10×10 = 105

or 103×102 = 103+2 = 105

a2×a5 = a×a × a×a×a×a×a = a7

or a2×a5 = a2+5 = a7

7 102×104 8 33×32 9 24×23 10 105×103

11 x2×x3 12 x4×x2 13 4.13×4.13 14 d3×d5

15 x×x4 16 y3×y 17 0.23×0.24 18 a3×a2

19 2.3×2.35 20 102×103 21 103×105×102 22 x4×x2×x

Simplify and write the following in index form:

103÷102 = 2 2 22 2× ××

= 10

or 103÷102 = 103−2 = 10

a6÷a2 = a a a a a aa a

× × × × ×× = a×a×a×a = a4

or a6÷a2 = a6−2 = a4

23 104÷102 24 104÷103 25 46÷42 26 22÷22

27 x6÷x3 28 y4÷y2 29 106÷103 30 a4÷a3

31 105÷10 32 b5÷b3 33 35÷34 34 104÷1035 x8÷x3 36 4.35÷4.32 37 107×103÷105 38 y5÷y5

39 1010

5

3 40 xx

7

4 41 a aa

7 2

4×

42 10 1010 10

7 3

4 6×

×

Warmup 23 Index

BaseIndices save a lot of effort.

IndexLaw1

am×an = am+n

Index Law 2

am÷an = am−n

y = y1

10 = 101

m4÷m2 and mm

4

2

are the same thing.

3Chapter 1 Algebra 1

Exercise1.2Simplify and write the following in index form: (b4)2 = (b×b×b×b)2

= (b×b×b×b)×(b×b×b×b) = b8

or (b4)2 = b4×2 = b8

104×(102)3 = 104×106 = 1010

(b4)2b3 = b8×b3 = b11

1 (b2)4 2 (b2)3 3 (b3)2 4 (103)2

5 (x2)2 6 (x2)5 7 (y3)4 8 (y5)2

9 103(102)2 10 x5(x3)2 11 (23)225 12 b3(b3)5

Simplify each of the following:

100 = 1 h0 = 1 3×50 = 3×1 = 3 5b0 = 5×1 = 5

13 100 14 h0 15 x0 16 a0

17 5×100 18 5a0 19 4×30 20 2×10

21 (x0)2×x 22 b2(b0)3 23 10(105)0 24 10×(100)2

Write each of the following using a negative index:

1103 = 10−3

15b = b−5

110 = 10−1

110000 =

1104 = 10−4

25 1105 26

14b 27

110 28

1100

Simplify and write the following in index form: 102×10-3 = 102-3 = 10-1 10−3 ÷10−4 = 10-3- -4 = 10-3+4 = 10

29 10-3×102 30 10-2×104 31 105÷10-3 32 10-4÷10-2

33 5-2×53 34 10-2×106 35 x4÷x-2 36 10-2÷104

37 x-5×x4×x3 38 y4×y-7×y2 39 10-5÷103 40 y-4÷y-5

(10−2)4 = 10ˉ2×4 = 10−8 9(100)-3 = 9×100×-3 = 9×1 = 9

41 (10−2)4 42 (2−3)5 43 (a2)−3 44 (10−5)−2

45 (y2)4 46 2(x−3)0 47 (y4)−4 48 (y−1)−7

Index Law 3

(am)n = am×n

Zero Index

a0 = 1

NegativeIndex

a−m = 1am

4

Exercise1.3Expand each of the following: 4(a + 3) = 4a + 12

ˉa(a + 3) = ˉa2 − 3a

5x2(3x − 2y) = 15x3 – 10x2y

3(2b − 5) = 6b – 15

ˉb2(2b − 5) = ˉ2b3 + 5b2

ˉ5x3(3x2 − 2y3) = ˉ15x5 + 10x3y3

1 5(x + 2) 2 4(a + 3) 3 y(y + 2)4 −x(x + 2) 5 x2(3x − 3) 6 4x(2x − 5)7 −5x2(3x − 2) 8 −2y(2y2 − 1) 9 4a2(3a3 + 2b)

Simplify each of the following by expanding and then collecting like terms: 4(x + 3) − 3(x + 4) = 4x + 12 − 3x − 12 = x

–x(x − 1) − x(x − 4) = –x2 + x − x2 + 4x = ˉ2x2 + 5x

10 3(x + 5) − 2(x + 3) 11 −x(x − 1) − x(x − 2)12 2(x + 5) − 3(x + 1) 13 −a(a + 2) − a(a + 6)14 2(x + 5) − 3(x + 2) 15 −a2(a + 2) − a2(a + 6)

(x + 5)(x + 4) = x(x + 4) + 5(x + 4) = x2 + 4x + 5x + 20 = x2 + 9x + 20

(x3 + 5)(x2 – 3) = x3(x2 – 3) + 5(x2 – 3) = x5 – 3x3 + 5x2 – 15

16 (x + 2)(x + 1) 17 (x3 + 2)(x2 − 1)18 (x + 3)(x + 1) 19 (x2 + 3)(x3 − 1)20 (x + 3)(x + 1) 21 (x3 + 3)(x3 − 1)

(x + 3)2 = (x + 3)(x + 3) = x(x + 3) + 3(x + 3) = x2 + 3x + 3x + 9 = x2 + 6x + 9

(x2 – 4)2 = (x2 – 4)(x2 – 4) = x2(x2 – 4) – 4(x2 – 4) = x4 – 4x2 – 4x2 + 16 = x4 – 8x2 + 16

22 (x + 1)2 23 (x − 1)2

24 (x2 + 2)2 25 (x − 2)2

26 (x + 3)2 27 (x2 – 3)2

28 (x + 3)2 29 (x3 – 3)2

DistributiveLaw

Multiply each inside term by the outside term.

Distribute - to spread out, to cover everything.

+ times + = ++ times − = −− times + = −− times − = +

a(b + c) = ab + ac

5Chapter 1 Algebra 1

Exercise1.4Factorise each of the following:

6x + 9 = 3×2x + 3×3 = 3(2x + 3)

4xy − 6x = 2x×2y − 2x×3 = 2x(2y − 3)

10x2 − 8x = 2x×5x − 2x×4 = 2x(5x − 4)

1 6a + 9 2 4ab − 6a 3 10c2 − 8c4 14x + 10 5 4ab − 6b 6 8d2 − 6d7 9c + 12 8 8xy + 10x 9 16x2 − 12x10 6x + 10 11 12st − 15t 12 15p5 − 36p3

18 − 20a4

= 2×9 − 2×10a4

= 2(9 − 10a4)

4x5 − 10x = 2x×2x4 − 2x×5 = 2x(2x4 − 5)

8x5 − 12x3

= 4x3×2x2 − 4x3×3 = 4x3(2x2 − 3)

13 6 − 9b3 14 4x4 − 6x 15 10c4 − 8c5

16 4 + 10a4 17 4x5 + 8x 18 9d5 + 6d3

19 9 − 12x5 20 8y3 − 10y 21 9x2 − 12x3

22 6 + 10y2 23 12x2 + 15x 24 12y5 − 36y3

25 10a3 + 15a2 26 21x3 + 18x2 27 24a3 − 27a2

x(x − 5) + 4(x − 5) = (x − 5)(x + 4)

x(x − 2) − 3(x − 2) = (x − 2)(x − 3)

28 x(x + 5) + 3(x + 5) 29 x(x + 5) − 4(x + 5)30 x(x − 1) + 4(x − 1) 31 x(x − 1) − 2(x − 1)32 x(x − 6) + 3(x − 6) 33 x(x − 5) − 4(x − 5)34 x(x − 2) + 5(x − 2) 35 x(x − 3) − 7(x − 3)

The common term, a+b, is taken out and put at the front.

Factorisation

Distribute

a(b + c) ab + ac

Factorise

c(a + b) + d(a + b) = (a + b)(c + d)

Factorisation is the inverse of distribution.

Algebra is an essential tool in thousands of careers and is fundamental to solving millions of problems.

6

Exercise1.5Simplify the following algebraic expressions:7x × 2x = 7 × 2 × x × x = 14x2

5ab2 × ˉ3a2b3 = 5 × ˉ3 × a × a2 × b2 × b3

= ˉ15a3b5

1 5x × 2x 2 3y × 4y3 3a × 2a 4 8b × 3b5 e7 × 2e3 6 8x3 × 2x5

7 5x2 × −2x 8 3x × −4x9 −5x × 4x3 10 −4x2 × −2x11 6m × −2m3 12 −5w × 3w4

13 −4h2 × −4h 14 −3x5 × 5x15 −3p2d × −2pd 16 4ab × −2a2b3

17 7de × −4d2e 18 5mn × −3m2n19 −4a2b2c × −5a2bc 20 5x4y3z5 × −3x2yz3

−8x3 × 3x−2 = −8 × 3 × x3 × x−2

= −24x5a3b−2 × ˉ3ab3 = 5 × ˉ3 × a3 × a × b−2 × b3

= ˉ15a4b

21 10x4 × 2x−2 22 5a−2 × 4a4

23 3y5 × 3y−2 24 2p−6 × 7p3

25 4x6 × −2x−4 26 −2x−7 × 3x4

27 4x4y6 × 7x−2y−2 28 2a3b5 × 4a−2b−3

29 −6x−3y2 × 2x6y−3 30 −2e5f3 × −2e−2f−1

31 −3a−2b6 × 7a4b−3 32 −2a−2b−2 × −2a4b3

4x−5 × 2x3 = 4 × 2 × x−5 × x3

= 8x−2

−3a−3b−2 × 2ab−3 = −3 × 2 × a−3 × a × b−2 × b−3

= ˉ6a−2b−5

33 3x−4 × 4x2 34 −8x × 4x−6

35 6y−2 × −3y−1 36 −10d−3 × −2d5

37 −3g3 × 2g−4 38 −2a−6 × −4a2

39 4x−4y−6 × 3x−2y−2 40 2a3b−5 × 4a−2b−3

41 −5a−3b2 × 3a−6b−3 42 −2x−5y3 × −2x−2y−4

43 −5e−2f−6 × 2e4f−3 44 −4a−5b−3 × −3a4b−3

5a means 5 multiplied by a• 5a is a product• 5 is a factor• a is a factor

6p × −2p3 means 6p multiplied by −2p3

• 6p × −2p3 is a product• 6p is a factor• −2p3 is a factor

+ times − = −− times + = −− times − = +

Multiply the numbers.Multiply the letters.

IndexLaw1

am×an = am+n

SimplifyAlgebraicProducts

To simplify is to reduce to a simpler form.

6p × −2p3 = −12p4

Zero Index

a0 = 1

NegativeIndex

a−m = 1am

a = a1

x = x1

7Chapter 1 Algebra 1

Exercise1.6Simplify the following algebraic expressions:8x ÷ 4 = 2x

12x5 ÷ 4x2 = 3x5−2

= 3x3

−6x5 ÷ 4x2y = −64

5

2xx y

= −32

3xy

1 10x ÷ 5 2 16a ÷ 43 12x ÷ 3 4 14d ÷ 75 −8x ÷ 4 6 6x ÷ −37 −10y ÷ −2 8 −10a ÷ −29 14x6 ÷ 2x3 10 21x7 ÷ 3x4

11 8x4 ÷ 4x2 12 −4g3 ÷ 2g2

13 −12a6 ÷ −4a2 14 8x6 ÷ −4x4

15 16x5y ÷ −4x3 16 −14x4y ÷ −7x2

17 −20ab ÷ 4b 18 −16a9c2 ÷ 12a6

19 −24e5f3 ÷ −12e2 20 −21a5b6c ÷ 28a4b3

6x−3 ÷ 2x2 = 3x−3−2

= 3x−5

−4x2y−4 ÷ 2xy−3 = −2x2−1y−4−−3 {x = x1} = −2x1y−4+3 {− −3 = 3} = −2xy−1

21 6a−3 ÷ 3a2 22 6b3 ÷ 2b−1

23 42

3

2xx

−

24 123

5

2aa−

25 −9w ÷ 3w−2 26 8s−3 ÷ 2s−2

27 15x2 ÷ 5x−4 28 3y−2 ÷ 2y

29 186

2

3xx

−

− 30 147

5

2nn

−

31 12m−2n ÷ −3m2n 32 8ab−3 ÷ −4a−2b33 4ab−1 ÷ −2a−2b4 34 −8x2y−1 ÷ −2xy−1

35 102

3 2

1 2m nm n

−

− − 36

−

− − −153

2

2 3ab

a b37 −4c−2d2 ÷ 4c2d2 38 −4a2b2c ÷ −a−2bc−2

Index Law 2

am÷an = am−n

Divide the numbers.Divide the letters

+ divided by − = −− divided by + = −− divided by − = +

8a÷2 means 8a divided by 2• 8a is the dividend• 2 is the divisor• 4a is the quotient

6p ÷ −2p3 means 6p divided by −2p3

• 6p is the dividend• −2p3 is the divisor• −3p−2 is the quotient

SimplifyAlgebraicQuotients

To simplify is to reduce to a simpler form.

6p ÷ −2p3 = −3p−2

Zero Index

a0 = 1

NegativeIndex

a−m = 1am

x1 = x

A quotient is the result of division.

8

Exercise1.7Simplify the following algebraic expressions:

1 x x5

25

+ 2 34

64

a a+

3 53

23

b b+ 4

c c6

46

+

5 43 3x x+ 6

45

25

2 2x x+

7 34

54

e e+ 8

34

54

3 3a a+

9 53

43

a a+ 10

37 7

3 2x x+

11 43 3x x+ 12

26

76

5 2y y+

13 35 5x x− 14

32 2a a−

15 73

23

y y− 16

46

36

a a−

17 53

23

c c− 18

23 3e e−

19 74

34

x x− 20

54

34

3 3x x−

21 92

52

y y− 22

75 5

2 2x y−

23 78 8z z− 24

33

23

5 2x x−

OperationswithAlgebraicFractions

ab

cb

a cb

+ =+

ab

cb

a cb

− =−−

27

37

x x+

= 2 37

x x+

= 57x

59

39

x x−

= 5 39

x x−

= 29x

34

74

x x+

= 3 74

x x+

= 104x

= 52x

94

34

x x−

= 9 34

x x−

= 64x

= 32x

38

28

5 5x x+

= 3 2

8

5 5x x+

= 58

5x

45

25

2 2x x−

= 4 2

5

2 2x x−

= 25

2x

+

3x5−2x2 = 3x5−2x2 The terms are not the same - they can't be subtracted.

3x3+x2 = 3x3+x2 The terms are not the same - they can't be added.

To subtract fractions, the denominator needs to be the same.

To add fractions, the denominator needs to be the same.

9Chapter 1 Algebra 1

Exercise1.8Simplify the following algebraic expressions:

1 3 4x x× 2

5 22a a

×

3 y y2 3× 4

y y3

4 3×

5 x x×5 6

45

3x x×

7 97

23

3x x× 8

45

32

4x×

9 79

62

3a a× 10 45 6

3x x×

11 23

14

6

2x

x× 12

56

315

3

5x

x×

13 x x2 3÷ 14

a a3 2÷

15 m m4 3÷ 16

25

34

x x÷

17 43

23

e÷ 18

67

45

x÷

19 32

25

4x x÷ 20

35

34

3 2y y÷

21 42

12

5t÷ 22

32

22

3ab

ab

÷

23 6 42

2xy

xy

÷ 24 127

82 3a bc

abc

÷

OperationswithAlgebraicFractions

ab

cd

acbd

× =

ab

cd

ab

dc

÷ = ×

×

÷

25

34a a×

= 2 35 4××a a

= 620 2a =

310 2a

23

45

x x÷

= 23

54

xx

×

= 1012xx =

56

67

58

5

3x

x×

= 6 57 8

5

3xx×

×

= 3056

5

3xx

= 1528

2x

65

32

5 3x x÷

= 65

23

5

3x

x×

= 1215

5

3xx =

45

2x

Multiply the numerators.Multiply the denominators.

In division, the second fraction is turned upside.

Multiply the numbers.Multiply the letters.

IndexLaw1

am×an = am+n

Index Law 2

am÷an = am−n

3 31

x x=

10

MentalComputation

Exercise1.91 Spell Quotient2 5 − 73 3 − ˉ44 102×103

5 x3 ÷ x2

6 (2−3)2

7 Simplify: x x2 3+

8 Simplify: x x2 3−

9 Increase $6 by 10%10 If I paid $50 deposit and 10 payments of $10. How much did I pay?

Exercise1.101 Spell Distribution2 ˉ2 − 33 3 × ˉ44 105×10ˉ3

5 x5 ÷ x2

6 (23)ˉ2

7 Simplify: x x2 5+

8 Simplify: x x2 5−

9 Increase $8 by 10%10 If I paid $50 deposit and 10 payments of $15. How much did I pay?

Exercise1.111 Spell Factorisation2 ˉ5 − 13 1 ÷ ˉ24 10ˉ2×103

5 xˉ3 ÷ x7

6 (2−3)ˉ2

7 Simplify: x x3 5+

8 Simplify: x x3 5−

9 Increase $9 by 10%10 If I paid $100 deposit and 10 payments of $25. How much did I pay?

You need to be a good mental athlete because many everyday problems are solved mentally.

Algebra is the greatest labour saving device ever invented by humans.

ab

cd

ad bcbd

+ =+ a

bcd

ad bcbd

− =−

10% of $6 is $0.60

'If you think dogs can't count, try putting three dog biscuits in your pocket and then giving Fido only two of them' - Phil Pastoret.

Stockbrokers buy and sell shares and bonds for clients.• Relevant school subjects are Mathematics and English.• Courses usually involve a Universtity Bachelor degree with a

major in commerce/finance.

11Chapter 1 Algebra 1

CompetitionQuestionsBuild maths muscle and prepare for mathematics competitions at the same time.

Exercise1.121 Put the following fractions in order of increasing size:

2 Evaluate each of the following: a) 1 + 2 × 3 − 4 b) 12 ÷ 3 × 4 − 6 + 7 c) 9 × 8 ÷ 6 − 5 d) (10 + 2) × 5 − 5 e) ((((1 − 2) − 3) − 4) − 5) f) 6 − (5 − (4 − (3 − (2 − 1))))

3 Simplify each of the following: a) 105×102

b) 105÷103

c) 102÷103×104

d) 107÷109×102

4 Simplify each of the following: a) 7x + 2y − 3x + 4y b) 4x − 2y − 8x + y c) (2a + b) − (a + 5b) d) 3(x − 2) − 3(x − 5) e) (x − 2) − (1 − x) f) 2x(x − 1) + 5x2

5 If 2(x+3) = 32, what is the value of x?

6 If 4(2x−1) = 16, what is the value of x?

7 Simplify: 1 1 1a b c+ +

8 If 1 1

223x

= + , what is the value of x?

OrderofOperations:1 ( ) brackets first.2 × and ÷ from left to right.3 + and − from left to right.

107÷105

= 107−5

= 102 or 100

2(x − 1) − 3(x − 4)

= 2x − 2 − 3x + 12 = ˉx + 10

32 = 25

12

13

15

+ +

= 3 52 3 5

2 53 2 5

2 35 2 3

×× ×

+×× ×

+×× ×

= 1530

1030

630

+ +

= 3130

143

4 13

23

12

12

Investigation1.1 Shortcutforsquaringnumberslessthan10

Complete the pattern: 22 = 3×1 + 1 62 = 32 = 4×2 + 1 72 = 42 = 5×3 + 1 82 = 52 = 6×4 + 1 92 =

Investigation1.2 Shortcutforsquaringnumbersnear10

Example: 132 {13 = 10 + 3}

132 = 102 + 20×3 + 32

= 100 + 60 + 9 = 169

Example: 162 {16 = 10 + 6}

162 = 102 + 20×6 + 62

= 100 + 120 + 36 = 256

Investigation1.3 Shortcutforsquaringnumbersnear50

Example: 532 {53 = 50 + 3}

532 = 502 + 100×3 + 32

= 2500 + 300 + 9 = 2809

Example: 462 {46 = 50 − 4}

462 = 502 + 100×−4 + −42

= 2500 + −400 + 16 = 2116

Investigation1.4

What is 152?What is 142?What is 172?

Investigations

What is 552?What is 542?What is 472?

What is 92? {9 = 10 − 1} 92 = 102 + 20×−1 + −12

= 100 + −20 + 1 = 81

(10 + a)2 = (10 + a)(10 + a) = 10(10 + a) + a(10 + a) = 102 + 10a +10a + a2

= 100 + 20a + a2

(10 + 3)2 = (10 + 3)(10 + 3) = 10(10 + 3) + 3(10 + 3) = 102 + 10×3 +10×3 + 32

= 100 + 60 + 9 = 169

(50 + 3)2 = (50 + 3)(50 + 3) = 50(50 + 3) + 3(50 + 3) = 502 + 50×3 +50×3 + 32

= 2500 + 300 + 9 = 2809

(50 + a)2 = (50 + a)(50 + a) = 50(50 + a) + a(50 + a) = 502 + 50a +50a + a2

= 2500 + 100a + a2

Investigatesquaringnumbersnear100?

13Chapter 1 Algebra 1

Exercise1.131 A cup and saucer together weigh 360 g. If the cup weighs twice as much as the saucer, what is the weight of the saucer?2 A cup and saucer together costs $25. If the cup cost $9 more than the saucer, what is the cost of the saucer?3 Complete the following multiplication problems: a) b) c) d)

Target is played by two people, or two teams, using four dice.

1 Take turns to choose a number from the board. This is then the target number.2 Roll the four dice. The first person, or team, to arrange the four numbers on the dice to equal the target number scores a hit.

Passaringthroughastring.1 Thread a light rope/string through two rings.2 Grip the two rings as shown in the photo.3 With flair, very quickly pull the ring in the right hand down the rope.4 Hey presto. The ring in the left hand is through the rope.

ACoupleofPuzzles

A Game

ASweetTrick

First practice the trick.Then add exaggerated gestures.

11 12 13

14 15 16

17 18 19

Example:Target = 14Dice numbers = 2, 5, 3, 1

14 = 5×2+3+114 = 3×5+1−214 = 23+5+1

Is the game too easy?Change the numbers to 21 to 29. or 31 to 39

5 6 × 2 7 0 90110

2 3× 81 2 04 2 8

3 8 ×26 4 08 9

43 18 ×44 43074

14

Technology1.1 SimplifyingFractionsScientific calculators are excellent in working with fractions:

1 Simplify 1535 15 a

bc 35 = 3r7 meaning

37

2 Simplify 184 18 a bc 4 = 4r1r2 meaning 4 12

To change to a vulgar fraction: 2ndF a bc to give 9r2 ie 92

3 Use a scientific calculator to simplify the following ratios: a) 3 : 9 b) 9 : 12 c) 16 : 24 d) 2.1 : 3.5 e) 14.4 : 12.6 f) 256 : 1024

Technology1.2 ExpandingandFactorisingGraphics calculators are capable of expanding and factorising:1 Choose expand from the algebra menu.2 Enter the algebraic expression: 3(4x − 5) to produce 12x − 15

1 Choose factor from the algebra menu.2 Enter the algebraic expression: 12x − 15 to produce 3(4x − 5)

Technology1.3 TheDistributiveLawandFactorisingThere are a considerable number of resources about the Distributive Law and factorising on the Internet.

Try some of them.

Technology1.4 AlgebraicFractions

Technology

The human mind has never invented a labor-saving machine equal to algebra.

Algebraicfractions

Watch videos on adding, subtracting, multiplying, and dividing fractions'.

I'The essence of mathematics is not to make simple things complicated, but to make complicated things simple.' - S. Gudder.

15Chapter 1 Algebra 1

ChapterReview1

Exercise1.14Expand each of the following:4(a + 3) = 4a + 12

ˉa(a + 3) = ˉa2 − 3a

ˉb2(2b − 5) = ˉ2b3 + 5b2

(x + 5)(x + 4) = x(x + 4) + 5(x + 4) = x2 + 4x + 5x + 20 = x2 + 9x + 20

(x + 3)2 = (x + 3)(x + 3) = x(x + 3) + 3(x + 3) = x2 + 3x + 3x + 9 = x2 + 6x + 9

1 5(x + 2) 2 −x(x + 2) 3 (x3 + 2)(x2 − 1)4 (x + 3)(x + 1) 5 (x + 3)2 6 (x3 + 3)(x3 − 1)

Factorise each of the following:

6x + 9 = 3×2x + 3×3 = 3(2x + 3)

4xy − 6x = 2x×2y − 2x×3 = 2x(2y − 3)

10x2 − 8x = 2x×5x − 2x×4 = 2x(5x − 4)

7 6x + 10 8 12st − 15t 9 15p5 − 36p3

10 6 + 10y2 11 12x2 + 15x 12 12y5 − 36y3

Simplify the following algebraic expressions:7x × −2x = 7 × −2 × x × x = −14x2

5a3b−2 × ˉ3ab3 = 5 × ˉ3 × a3 × a × b−2 × b3

= ˉ15a4b

13 6m × −2m3 14 −5w × 3w4 15 −4h2 × −4h16 10x4 × 2x−2 17 5a−2 × 4a4 18 3y5 × 3y−2

19 4x4y6 × 7x−2y−2 20 2a3b5 × 4a−2b−3 21 −6x−3y2 × 2x6y−3

6x−3 ÷ 2x2 = 3x−3−2

= 3x−5−4x2y−4 ÷ 2xy−3 = −2x2−1y−4−−3 {x = x1} = −2x1y−4+3 {− −3 = 3} = −2xy−1

22 14x6 ÷ 2x3 23 −12a6 ÷ −4a2 24 8x6 ÷ −4x4

25 −9w ÷ 3w−2 26 8s−3 ÷ 2s−2 27 4ab−1 ÷ −2a−2b4

28 186

2

3xx

−

− 29 10

2

3 2

1 2m nm n

−

− − 30

−

− − −153

2

2 3ab

a b

31 43 3x x+ 32

45

25

2 2x x+

33 34

54

e e+ 34

34

54

3 3a a+

35 74

34

x x− 36

54

34

3 3x x−

37 92

52

y y− 38

75 5

2 2x y−

38

28

5 5x x+

= 3 2

8

5 5x x+

= 58

5x

45

25

2 2x x−

= 4 2

5

2 2x x−

= 25

2x

16

ChapterReview2

Exercise1.15Expand each of the following:4(a + 3) = 4a + 12

ˉa(a + 3) = ˉa2 − 3a

ˉb2(2b − 5) = ˉ2b3 + 5b2

(x + 5)(x + 4) = x(x + 4) + 5(x + 4) = x2 + 4x + 5x + 20 = x2 + 9x + 20

(x + 3)2 = (x + 3)(x + 3) = x(x + 3) + 3(x + 3) = x2 + 3x + 3x + 9 = x2 + 6x + 9

1 3(x + 4) 2 −a(a + 5) 3 (x2 + 4)(x3 − 1)4 (x + 2)(x + 1) 5 (x + 2)2 6 (x2 + 3)(x2 − 1)

Factorise each of the following:

6x + 9 = 3×2x + 3×3 = 3(2x + 3)

4xy − 6x = 2x×2y − 2x×3 = 2x(2y − 3)

10x2 − 8x = 2x×5x − 2x×4 = 2x(5x − 4)

7 6x + 8 8 12ab − 9a 9 9x5 − 15x2

10 8 + 12y2 11 6x3 + 15x 12 18y4 − 24y3

Simplify the following algebraic expressions:7x × −2x = 7 × −2 × x × x = −14x2

5a3b−2 × ˉ3ab3 = 5 × ˉ3 × a3 × a × b−2 × b3

= ˉ15a4b

13 5x × −2x4 14 −5x × 4x3 15 −2y5 × −3y16 7x5 × 2x−2 17 3a−2 × 4a5 18 2y6 × 3y−2

19 3x5y3 × 6x−3y−3 20 5x3y4 × 4x−2y−1 21 −4a−2b4 × 2a5b−3

6x−3 ÷ 2x2 = 3x−3−2

= 3x−5−4x2y−4 ÷ 2xy−3 = −2x2−1y−4−−3 {x = x1} = −2x1y−4+3 {− −3 = 3} = −2xy−1

22 10x5 ÷ 2x3 23 −10y7 ÷ −4y3 24 6x5 ÷ −4x4

25 −15b2 ÷ 3b−3 26 8c−4 ÷ 6c−2 27 10ab−3 ÷ −2a−2b3

28 1518

4

2xx

− 29

1410

5 3

2 2a ba b

−

− 30 − −

− −146

2

2 3xyx y

31 x x5

25

+ 32 34

64

a a+

33 43 3x x+ 34

45

25

2 2x x+

35 35 5x x− 36

32 2a a−

37 74

34

x x− 38

54

34

3 3x x−

38

28

5 5x x+

= 3 2

8

5 5x x+

= 58

5x

45

25

2 2x x−

= 4 2

5

2 2x x−

= 25

2x