Complex Rationals - Intermediate Algebra - Lecture Slides, Slides of Algebra

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§6.3 Complex

Rational Fcns

Review §

 Any QUESTIONS About

• §6.2 → Add-n-Sub Rational

Expressions

 Any QUESTIONS About HomeWork

• §6.2 → HW-

MTH 55

Simplify Complex Rational Expressions

by Dividing

1. Add or subtract, as needed, to get a single

rational expression in the numerator.

2. Add or subtract, as needed, to get a single

rational expression in the denominator.

3. Divide the numerator by the denominator

( invert and multiply ).

4. If possible, simplify by removing any

factors equal to 1

Example  Simplify

 SOLUTION

4 3 2 8

x x

x

−

−

4 3 (^3 4 2 ) 2 8

x x x x x x

− (^) = ÷ − − − 2 8 4 3

x x x

= ⋅^ − −

4

x x

= −

⋅ 2(^ x^ −^4 ) 3 2 3

=^ x

Rewriting with a division symbol

Multiplying by the reciprocal of the divisor (inverting and multiplying)

Factoring and removing a factor equal to 1.

Solution cont.

5 3 4 1 2

x

x x

− 15 15 3 3 3 8 1 7 2 2 2

x x

x x x

+^ + = = −

15 7 3 2

x x

= ÷

15 2 3 7

• x x = ⋅

30 2 2 21

x + x

Rewriting with a division symbol. This is often done mentally.

Multiplying by the reciprocal of the divisor (inverting and multiplying)

Add and Subtract Using COMMON Denominators

Example  Simplify

2 1

3 2

−

y

 SOLUTION y

Write the numerator and denominator

as equivalent fractions.

2 1

3 2

−

y

y

1 1

2 1 1 3 2 1

1 y y 1

y y y y

⋅ + ⋅

⋅ − ⋅

2 1

3 2

= (^) −

y y y y

= 2^ y^ +^1 ÷^3 y −^2 y y

2 1 3 2

= + ⋅ −

y y y y

2 1 3 2

y y

= + −

2 1

3 2

= −

y y y y y y

Example  Simplify

 SOLUTION - In This Case look

for the LCD of all four Terms.

1 3 2 4 2 1 3 4

12 1

1 3 1 3 2 4 2 4 2 1 2 3 4 3 4

(^12)

= ⋅

12 12 12

1 3 1 3 2 4 2 4 (^2 1 2 ) 3 4 3 4 12

• ^ +    ⋅ =
• ^ +   

Multiplying by a factor equal to 1, using the LCD: 12/12=

Multiplying the numerator by 12 Don’t forget the parentheses! Multiplying the denominator by 12

Solution cont.

1 3 2 4 2 1 3 4

12 12 12

1 3 1 3 2 4 2 4 (^2 1 2 ) (^3 )

2 (^4 )

1

 (^) + 

• (^)   ⋅ =
• ^ +   

(^1) ( ) 3 ( ) 2 4 (^2) ( ) 1

12 12

12 ( 12 ) 3 4

=

6 9 , or 15

Using the distributive law

Simplifying

Example  Simplify

 SOLUTION

3 2

2

6 2

3 4

x x

x x

−

3 2 3 2

2 2

3 3

6 2 6 2

3 4 3 4

x x x x

x

x

x

x x x

− − = ⋅

The LCD for all is x^3 so we multiply by 1 using x^3 / x^3.

3 3 3 2 3 2

3

6 2

3 4

x x

x

x x

x x x

⋅ − ⋅

⋅ + ⋅

2

x x

x x x x

= −^ = −

Using the distributive law

All fractions have been cleared and simplified.

Example  Simplify

 SOLUTION: Multiply the

numerator and denominator by the LCD

of all the rational expressions; 2 x here

3 2 1 1

−

x x x x 3 2

1 1

−

x x x x

3

1 1

2 2 2

x x

x

x

x x

 (^) − ⋅   =  (^) + + ⋅  

2 3 2 2 1 1 1 2 1 2 1 1 1

⋅ − ⋅ = (^) + ⋅ + ⋅

x x x x x x x x (^2 ) 2 2( 1)

= −

x x x

(^2 ) 2 2 2

= −

x x x

(^2 6 ) or 4 2 2(2 1)

= −^ −

x x x x

Example  Simplify

 SOLN

cont.

2

3 2

5 2

1 5.

x xy y

y x y

−

2 3 2 2 3

3 2 3 2 2 3

5 2

1 5

x xy y

y

x y x y

x y (^) x y x y

⋅ + ⋅

⋅ − ⋅

Using the distributive law to carry out the multiplications

2 2 2 2 2

5 2 5

x y x y x y = + − 2 2 2

(5 2) 5

x y y x y = + −

Removing factors that equal

1. Study this carefully. Take CARE with CANCELLING

Simplifying

Factoring. This does not simplify further

2 2 2 2 2 (^3 2 ) 3 2

5 2

5

xy (^) x y y x y xy y y (^) x x y y y x y

⋅ + ⋅

⋅ − ⋅

Example  Simplify

 SOLUTION:

Rewrite using only positive exponents

2 2 3 3

x y x y

− − − −

−

2 2 3 3

x y x y

− − − −

−

2 2

3 3

1 1

1 1

x y

x y

= −

3 3 3 3

xy x y y x

= + −

2 2 2 2

( ) ( )( )

xy y x y x y xy x

= + − + +

3 3 2

3

3 3

3

2

1 1

1 1

x y

x y

x y

x y

 (^) + ⋅   = ^   (^) − ⋅    

LCD of all individual Rational Expressions is x^3 y^3

Simplified Version is still a bit “Complex”

All Done for Today

More Info

on

LCDs

 L iquid C rystal D isplay

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

Chabot Mathematics

Appendix

r − s ≡^ (^ r − s )( r^ + s )

2 2