Chapter 5: Algebra 1

SET 1

Chapter 5

Algebra

رــجبال

2 Chapter 5: Algebra

5.1 Basic Operations العمليات األساسية

Chapter 5: Algebra 3

4 Chapter 5: Algebra

5.2 Laws of Indices قوانيـن األسـس

Chapter 5: Algebra 5

5.3 Brackets and Factorisation

رفع األقواس و التحليل الى العوامل

6 Chapter 5: Algebra

Chapter 5: Algebra 7

5.4 Order of Operations تنفيذ العمليات تسلسل

8 Chapter 5: Algebra

Chapter 5: Algebra 9

5.5 Polynomials كثيـزات الحـذود

A polynomial is an algebraic expression in which all terms have variables that are raised to

whole number powers.

Polynomial terms cannot contain variables which are raised to fractional powers or terms

which contains variables in the denominator.

All the following expressions are polynomials:

435 2 xx

846 332 xyyxyx

45z

12

zxy 42

While the following expressions are not polynomials:

xx

323 4

843 22 xxyyx

34 324 xx

The degree of a polynomial is equal to the degree of the term having the highest degree.

For example, the degree of the first term of is 6 which is higher

than the degree of each of the other three terms and hence the degree of this

polynomial is 6.

Example 73: Determine whether each of the following expressions is a polynomial or not:

(a) 4113 2 xx (b) 42 78 xyxzx (c) 282 xyy

(d) 16

(e) 22 24

xyz

x (f) 542 32 xy

Solution:

(a) 4113 2 xx is a polynomial

(b) 42 78 xyxzx is a polynomial

(c) 282 xyy is not a polynomial

(d) 16 is a polynomial

(e) 22 24

xyz

x is not a polynomial

(f) 542 32 xy is not a polynomial

532 2342 yxyxyx

10 Chapter 5: Algebra

Example 73: Determine the degree of each of the following polynomials:

(a) 222

876 zxzx (b) 2225 53 yxxx (c) y6

(d) 8 and (e) xyz6

Solution:

(a) 222876 zxzx is a fourth degree polynomial

(b) 2225 53 yxxx is a fifth degree polynomial

(c) y6 is a first degree polynomial

(d) 8 is a zero degree polynomial

(e) xyz6 is a third degree polynomial

Example 73: Find the sum of )373( 2 xx and )82( 23 xx .

Solution:

57

57

82373)82()373(

23

32

232232

xxx

xxx

xxxxxxxx

Example 40: Find the sum of )93613( 32 xyxz and )158511( 22 xyzx .

Solution:

65883

86358

15851193613)158511()93613(

223

232

22322232

xzxyxy

xyxyxz

xyzxxyxzxyzxxyxz

Example 41: Subtract )93613( 32 xyxz from )158511( 22 xyzx .

Solution:

24171883

32481817

93613158511)93613()158511(

223

322

32223222

xzxyxy

xyxyzx

xyxzxyzxxyxzxyzx

Chapter 5: Algebra 11

Example 42: Multiply )46( 23 xz by )427( 2 zxy .

Solution:

2223353223 16828241242)427)(46( xzxyxzzxyzzxyxz

Example 43: Divide )473( 2 xx by )1( x .

Solution:

In division of polynomials, long division is used in the same way it is used in the division of numbers.

The result of division of polynomials may or may not contain a remainder and as illustrated in this

example and the following examples.

Using long division:

43 x

47312

xxx

xx 332

44 x

44 x

0

Thus 43)1()473( 2 xxxx

Example 44: Determine )91144( 23 xxx )12( x .

Solution:

Using long division:

2x2 + 3x 4

9114412 23 xxxx

23 24 xx

9116 2 xx

xx 36 2

98 x 48 x

13

Thus 432)12()91144( 223 xxxxxx remainder 13 ,

or 12

13432)12()91144( 223

xxxxxxx

12 Chapter 5: Algebra

Example 45: Find: 2

18232 35

x

xxx

Solution:

Using long division:

x4 2x

3 2x

2 4x 31

18230202 2345 xxxxxx

45 2xx

1823022 234 xxxx

34 42 xx

182302 23 xxx

242 3 xx

18234 2 xx

xx 84 2

1831 x

6231 x

80

Hence 31422)2()18232( 23435 xxxxxxxx remainder 80 ,

or 2

8031422)2()18232( 23435

xxxxxxxxx

5.6 Rational Expressions التعابيـز الجبزية النسبيـة

Fractional expressions such as x

3 and 8

642

x

xx

are called rational expressions since they

have polynomials as both numerator and denominator.

A rational expression is proper if the degree of the numerator is less than the degree of the

denominator.

For example, 82 x

x

is a proper rational expression.

If the degree of the numerator is greater than or equal to the degree of the denominator, then

the rational expression is improper.

For example, 12

2

x

x and 1

123

x

xx are both improper rational expressions.

Chapter 5: Algebra 13

Example 46: Determine whether each of the following fractional expressions is a rational

expression or not. For rational expressions, determine whether they are proper

or improper.

(a) 56

33

2

x

x (b)

82 2

3

y

y (c)

1

522

x

x

Solution:

(a) 56

33

2

x

x is a proper rational expression.

(b) 82 2

3

y

y is an improper rational expression.

(c) 1

522

x

x is not rational expression.

Example 47: Simplify the following: (a) 52

3

4

2

xx (b) 432

x

C

x

B

x

A

Solution:

(a) )52)(4(

)4(3)52(2

52

3

4

2

xx

xx

xx

2032

22

)52)(4(

123104

2

xx

x

xx

xx

(b) )4)(3)(2(

)3)(2()4)(2()4)(3(

432

xxx

xxCxxBxxA

x

C

x

B

x

A

)4)(3)(2(

)6()86()12( 222

xxx

xxCxxBxxA

)4)(3)(2(

68612 222

xxx

CCxCxBBxBxAAxAx

)4)(3)(2(

68126222

xxx

CBACxBxAxCxBxAx

)4)(3)(2(

)6812()6()( 2

xxx

CBAxCBAxCBA

14 Chapter 5: Algebra

5.7 Rationalizing Denominators and Numerators التخلص من الجذور في البسط أو المقام

In working with quotients involving radicals, it is often convenient to move the radical

expression from the denominator to the numerator, or vice versa.

This process is called rationalizing the denominator or rationalizing the numerator.

Example 48: Rationalize the denominator or numerator and simplify:

(a) 2

1x (b)

8

4

y

y (c)

1

1

xx

Solution:

(a) 1

1

2

1

2

1

x

xxx

12

1

x

x

(b) 8

8

8

4

8

4

y

y

y

y

y

y

8

84

y

yy

(c) 1

1

1

1

1

1

xx

xx

xxxx

)1(

1

xx

xx

1

1

xx

xx

1

1

xx

1 xx