5.1 Basic Operations Chapter 5: Algebra 2 Chapter 5: Algebra - - PDF document

Chapter 5: Algebra

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SET 1

Chapter 5

Algebra

لابجــر

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Chapter 5: Algebra

5.1 Basic Operations ةيساسلؤا تايلمعلا

Chapter 5: Algebra

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Chapter 5: Algebra

5.2 Laws of Indices سـسلؤا نـيناوق

Chapter 5: Algebra

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5.3 Brackets and Factorisation لماوعلا ىلا ليلحتلا و ساوقلؤا عفر

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Chapter 5: Algebra

Chapter 5: Algebra

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5.4 Order of Operations

لسلست تايلمعلا ذيفنت

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Chapter 5: Algebra

Chapter 5: Algebra

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5.5 Polynomials دوذـحلا تازـيثك

 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:

4 3 5 2   x x 8 4 6

3 3 2

   xy y x y x

4

5z 

12

z xy 4

2 

 While the following expressions are not polynomials:

x x 3 2 3 4  8 4 3

2 2

   x xy y x 3 4

3 2 4

  x x

 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) 4 11 3 2   x x (b)

4 2

7 8 xy x zx   (c) 2 8 2    x y y (d) 16 (e)

2 2

2 4 xy z x   (f)

5 4 2

3 2 x y 

Solution:

(a) 4 11 3 2   x x is a polynomial (b)

4 2

7 8 xy x zx  

is a polynomial (c) 2 8 2    x y y

is not a polynomial

(d) 16 is a polynomial (e)

2 2

2 4 xy z x   is not a polynomial (f)

5 4 2

3 2 x y  is not a polynomial

5 3 2

2 3 4 2

   y x y x y x

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Chapter 5: Algebra

Example 73: Determine the degree of each of the following polynomials: (a)

2 2 2

8 7 6 z x z x  

(b)

2 2 2 5

5 3 y x x x  

(c) y 6 (d) 8 and (e) xyz 6

Solution:

(a)

2 2 2

8 7 6 z x z x  

is a fourth degree polynomial (b)

2 2 2 5

5 3 y x x x  

is a fifth degree polynomial (c) y 6 is a first degree polynomial (d) 8 is a zero degree polynomial (e) xyz 6 is a third degree polynomial

Example 73: Find the sum of

) 3 7 3 (

2

  x x

and ) 8 2 (

2 3

  x x . Solution:

5 7 5 7 8 2 3 7 3 ) 8 2 ( ) 3 7 3 (

2 3 3 2 2 3 2 2 3 2

                   x x x x x x x x x x x x x x Example 40: Find the sum of

) 9 3 6 13 (

3 2

   xy x z

and ) 15 8 5 11 (

2 2

   xy z x . Solution:

6 5 8 8 3 8 6 3 5 8 15 8 5 11 9 3 6 13 ) 15 8 5 11 ( ) 9 3 6 13 (

2 2 3 2 3 2 2 2 3 2 2 2 3 2

                      

 

x z xy xy xy xy x z xy z x xy x z xy z x xy x z Example 41: Subtract

) 9 3 6 13 (

3 2

   xy x z

from ) 15 8 5 11 (

2 2

   xy z x . Solution:

24 17 18 8 3 3 24 8 18 17 9 3 6 13 15 8 5 11 ) 9 3 6 13 ( ) 15 8 5 11 (

2 2 3 3 2 2 3 2 2 2 3 2 2 2

                      

  

x z xy xy xy xy z x xy x z xy z x xy x z xy z x

Chapter 5: Algebra

11 Example 42: Multiply

) 4 6 (

2 3

x z 

by ) 4 2 7 (

2 

 z xy . Solution:

2 2 2 3 3 5 3 2 2 3

16 8 28 24 12 42 ) 4 2 7 )( 4 6 ( x z x y x z z xyz z xy x z          Example 43: Divide

) 4 7 3 (

2

  x x

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: 4 3  x

4 7 3 1

2

   x x x x x 3 3

2 

4 4   x 4 4   x Thus

4 3 ) 1 ( ) 4 7 3 (

2

      x x x x

Example 44: Determine ) 9 11 4 4 (

2 3

   x x x  ) 1 2 (  x . Solution: Using long division: 2x2 + 3x  4 9 11 4 4 1 2

2 3

    x x x x

2 3

2 4 x x  9 11 6 2   x x x x 3 6 2 

9 8   x 4 8   x 13  Thus 4 3 2 ) 1 2 ( ) 9 11 4 4 (

2

2 3

        x x x x x x remainder 13  ,

• r

1 2 13 4 3 2 ) 1 2 ( ) 9 11 4 4 (

2

2 3

          x x x x x x x

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Chapter 5: Algebra

Example 45: Find:

2 18 23 2 3

5

    x x x x

Solution: Using long division: x4  2x3  2x2  4x  31

18 23 2 2

2 3 4 5

      x x x x x x

4 5

2x x 

18 23 2 2

2 3 4

     x x x x

3 4

4 2 x x   18 23 2

2

3

   x x x

2

4 2 3 x x  18 23 4 2    x x x x 8 4 2   18 31  x 62 31  x 80  Hence

31 4 2 2 ) 2 ( ) 18 23 2 (

2 3 4 3 5

          x x x x x x x x

remainder 80  ,

• r

2 80 31 4 2 2 ) 2 ( ) 18 23 2 (

2 3 4 3 5

            x x x x x x x x x

5.6 Rational Expressions ةـيبسنلا ةيزبجلا زـيباعتلا

 Fractional expressions such as x 3 and 8 6 4

2

   x x x 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, 8

2 

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,

1

2 2

 x x

and

1 1 2

3

   x x x

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

• r improper.

(a)

5 6 3

3 2

 x x

(b)

8 2 2

3

 y y

(c)

1 5 2

2 

 x x

Solution: (a)

5 6 3

3 2

 x x

is a proper rational expression. (b)

8 2 2

3

 y y

is an improper rational expression. (c)

1 5 2

2 

 x x

is not rational expression. Example 47: Simplify the following: (a)

5 2 3 4 2    x x

(b)

4 3 2      x C x B x A

Solution: (a)

) 5 2 )( 4 ( ) 4 ( 3 ) 5 2 ( 2 5 2 3 4 2          x x x x x x

20 3 2 22 ) 5 2 )( 4 ( 12 3 10 4

2

          x x x x x x x (b)

) 4 )( 3 )( 2 ( ) 3 )( 2 ( ) 4 )( 2 ( ) 4 )( 3 ( 4 3 2                  x x x x x C x x B x x A x C x B x A ) 4 )( 3 )( 2 ( ) 6 ( ) 8 6 ( ) 12 (

2 2 2

            x x x x x C x x B x x A ) 4 )( 3 )( 2 ( 6 8 6 12

2 2 2

            x x x C Cx Cx B Bx Bx A Ax Ax ) 4 )( 3 )( 2 ( 6 8 12 6

2 2 2

            x x x C B A Cx Bx Ax Cx Bx Ax ) 4 )( 3 )( 2 ( ) 6 8 12 ( ) 6 ( ) (

2

             x x x C B A x C B A x C B A

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Chapter 5: Algebra

5.7 Rationalizing Denominators and Numerators ماقملا وأ طسبلا يف روذجلا نم صلختلا

 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 1  x

(b)

8 4  y y

(c)

1 1   x x

Solution: (a)

1 1 2 1 2 1       x x x x 1 2 1    x x

(b)

8 8 8 4 8 4       y y y y y y 8 8 4    y y y

(c)

1 1 1 1 1 1           x x x x x x x x ) 1 ( 1      x x x x 1 1      x x x x 1 1     x x 1     x x