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Copyright © 2007 Pearson Education, Inc. Slide R-1
Rational Expressions and
Equations
Chapter 9
pg 470
Copyright © 2007 Pearson Education, Inc. Slide R-2
Review of Rational Expressions
• A rational expression is an expression...
Copyright © 2007 Pearson Education, Inc. Slide R-3
Domain of a Rational Expression
• The domain of a rational expression i...
Copyright © 2007 Pearson Education, Inc. Slide R-4
Domain of a Rational Expression
For example, to find the domain of
solv...
Copyright © 2007 Pearson Education, Inc. Slide R-5
Lowest Terms of a Rational
Expression
Fundamental Principle of Fraction...
Copyright © 2007 Pearson Education, Inc. Slide R-6
Writing Rational Expressions in
Lowest Terms
Example Write each rationa...
Copyright © 2007 Pearson Education, Inc. Slide R-7
Writing Rational Expressions in
Lowest Terms
Solution
(b)
by the fundam...
Copyright © 2007 Pearson Education, Inc. Slide R-8
Multiplying and Dividing Rational
Expressions
Multiplying and Dividing ...
Copyright © 2007 Pearson Education, Inc. Slide R-9
Multiplying and Dividing Rational
Expressions
Example Multiply or divid...
Copyright © 2007 Pearson Education, Inc. Slide R-10
Multiplying and Dividing Rational
Expressions
Solution (b)
2
3 2 4 3 2...
Copyright © 2007 Pearson Education, Inc. Slide R-11
Complex Fractions
• Complex fractions are those fractions whose
numera...
Copyright © 2007 Pearson Education, Inc. Slide R-12
Now you try!
• Pg 476- 477
• #’s 14-22 evens
• #’s 26-34 evens
• #’s 3...
Copyright © 2007 Pearson Education, Inc. Slide R-13
Adding and Subtracting Rational
Expressions
Adding and Subtracting Fra...
Copyright © 2007 Pearson Education, Inc. Slide R-14
Adding and Subtracting Rational
Expressions
Finding the Least Common D...
Copyright © 2007 Pearson Education, Inc. Slide R-15
Adding/Subtracting
• when we talk about CDs, we mean denominators
that...
Copyright © 2007 Pearson Education, Inc. Slide R-16
Adding and Subtracting Rational
Expressions
Example Add or subtract, a...
Copyright © 2007 Pearson Education, Inc. Slide R-17
Adding and Subtracting Rational
Expressions
Solution (a) The LCD is
Th...
Copyright © 2007 Pearson Education, Inc. Slide R-18
Adding and Subtracting Rational
Expressions
Solution (b)
2 2 2
2
2 2 2...
Copyright © 2007 Pearson Education, Inc. Slide R-19
Now you try!
• Pg 482- 483 #’s 26-36 evens
Copyright © 2007 Pearson Education, Inc. Slide R-20
Complex Fractions
• A complex fraction is any quotient of two rational...
Copyright © 2007 Pearson Education, Inc. Slide R-21
Simplifying Complex Fractions
Example Simplify
Solution
Multiply both ...
Copyright © 2007 Pearson Education, Inc. Slide R-22
Simplifying Complex Fractions
Solution
2 2
( 1) ( 1)
11 1
11 1
1 1 1 1...
Copyright © 2007 Pearson Education, Inc. Slide R-23
Now you try!
• Pg 483 #’s 40
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  1. 1. Copyright © 2007 Pearson Education, Inc. Slide R-1 Rational Expressions and Equations Chapter 9 pg 470
  2. 2. Copyright © 2007 Pearson Education, Inc. Slide R-2 Review of Rational Expressions • A rational expression is an expression that is the quotient of two polynomials. Examples include 2 2 6 ( 6)( 4) 2 7 4 , , 2 ( 2)( 4) 5 20 x x x p p x x x p p + + + + − + + + +
  3. 3. Copyright © 2007 Pearson Education, Inc. Slide R-3 Domain of a Rational Expression • The domain of a rational expression is the set of real numbers for which the expression is defined. • The domain consists of all real numbers except those that make the denominator 0.
  4. 4. Copyright © 2007 Pearson Education, Inc. Slide R-4 Domain of a Rational Expression For example, to find the domain of solve as follows, or or The domain is ( 6)( 4) ( 2)( 4) x x x x + + + + ( 2)( 4) 0x x+ + = 2 0x + = 4 0x + = 2x = − 4x = − { | 2, 4} .x x ≠ − −
  5. 5. Copyright © 2007 Pearson Education, Inc. Slide R-5 Lowest Terms of a Rational Expression Fundamental Principle of Fractions ( 0 , 0) ac a b c bc b = ≠ ≠
  6. 6. Copyright © 2007 Pearson Education, Inc. Slide R-6 Writing Rational Expressions in Lowest Terms Example Write each rational expression in lowest terms. (a) (b) Solution (a) by the fundamental principle, provided p is not 0 or –4. 2 2 2 7 4 5 20 p p p p + − + 2 2 2 7 4 (2 1) 2 1( 4) 5 20 5 5( 4) p p p p p p p p p p + + + − − − = = + 2 6 3 4 k k − −
  7. 7. Copyright © 2007 Pearson Education, Inc. Slide R-7 Writing Rational Expressions in Lowest Terms Solution (b) by the fundamental principle. 2 6 3 3 3 4 ( 2) ( 2 (2 ) (2 )( 1) ( 2) ( 2)( 1) ( ) 3 3 ( 2) 2 )( 1) 2) ( )(2 k k k k k k k k k k k k − − − − − − − − = = − + + − = = + + − −
  8. 8. Copyright © 2007 Pearson Education, Inc. Slide R-8 Multiplying and Dividing Rational Expressions Multiplying and Dividing Fractions For fractions and and ( 0 , 0), c b d d ≠ ≠ a b a c ac b d bd × = , if 0. a c a d c b d b c d ÷ = × ≠
  9. 9. Copyright © 2007 Pearson Education, Inc. Slide R-9 Multiplying and Dividing Rational Expressions Example Multiply or divide as indicated. (a) (b) Solution (a) 2 3 2 4 3 3 11 4 9 36 24 8 24 36 p p p p p p p + − + ÷ − − 2 5 2 27 9 8 y y × 2 5 3 3 2 2 2 5 22 2 27 2 27 3 9 8 9 8 3 4 9 4 9 y y y y y y y y × × × × × = = × × × × × =
  10. 10. Copyright © 2007 Pearson Education, Inc. Slide R-10 Multiplying and Dividing Rational Expressions Solution (b) 2 3 2 4 3 2 3 3 2 3 2 3 11 4 9 36 ( 4)(3 1) 9( 4) 24 8 24 36 8 (3 1) 12 (2 3) (12 )(2 3) 8 (9) 12 (2 3) 9 8 (2 3) ( 4) ( (3 1) (3 1 )) 6 4 p p p p p p p p p p p p p p pp p p p p p p p p p p + − + + − + ÷ = ÷ − − − − −− − − = + × − = + =
  11. 11. Copyright © 2007 Pearson Education, Inc. Slide R-11 Complex Fractions • Complex fractions are those fractions whose numerator & denominator both contain fractions.
  12. 12. Copyright © 2007 Pearson Education, Inc. Slide R-12 Now you try! • Pg 476- 477 • #’s 14-22 evens • #’s 26-34 evens • #’s 36, 37, 38
  13. 13. Copyright © 2007 Pearson Education, Inc. Slide R-13 Adding and Subtracting Rational Expressions Adding and Subtracting Fractions For fractions and and ( 0 , 0), c b d d ≠ ≠ a b a c ad bc b d bd + + = . a c ad bc b d bd − − = •Addition and subtraction are typically performed using the least common denominator.
  14. 14. Copyright © 2007 Pearson Education, Inc. Slide R-14 Adding and Subtracting Rational Expressions Finding the Least Common Denominator (LCD) 1. Write each denominator as a product of prime factors. 2. Form a product of all the different prime factors. Each factor should have as exponent the greatest exponent that appears on that factor.
  15. 15. Copyright © 2007 Pearson Education, Inc. Slide R-15 Adding/Subtracting • when we talk about CDs, we mean denominators that contain the same factors. • To find our CD, we will first factor the ones we have. • Then we will multiply each denominator by the factors it is missing to create a CD. • Remember, we must also multiply the numerator by that same factor.
  16. 16. Copyright © 2007 Pearson Education, Inc. Slide R-16 Adding and Subtracting Rational Expressions Example Add or subtract, as indicated. (a) (b) Solution (a) Step 1: Find the LCD 2 2 2 3 2 4 2 y y y y y y + − − − +2 5 1 9 6x x +
  17. 17. Copyright © 2007 Pearson Education, Inc. Slide R-17 Adding and Subtracting Rational Expressions Solution (a) The LCD is Then 1 2 2 2 2 3 18 .x x× × = 2 2 2 2 2 5 1 5 1 9 6 9 6 10 3 18 18 1 2 . 2 0 3 3 3 18 x xx x x x x x x x x × × + = + × × = + + =
  18. 18. Copyright © 2007 Pearson Education, Inc. Slide R-18 Adding and Subtracting Rational Expressions Solution (b) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 3 2 4 2 ( 1) 2( 1) ( 2) 3 ( 1) 2( 1) 2( 2) 3 2 2 4 3 2 ( 1) 2 ( 1) 2 2( 1 ( 1) 2 4 ) 2( 2 ( ) 1 1 ) y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y − − + + − = − − − + − − + × × = − − × − × + − + − − = − = − − − − + − = −
  19. 19. Copyright © 2007 Pearson Education, Inc. Slide R-19 Now you try! • Pg 482- 483 #’s 26-36 evens
  20. 20. Copyright © 2007 Pearson Education, Inc. Slide R-20 Complex Fractions • A complex fraction is any quotient of two rational expressions.
  21. 21. Copyright © 2007 Pearson Education, Inc. Slide R-21 Simplifying Complex Fractions Example Simplify Solution Multiply both numerator and denominator by the LCD of all the fractions a(a + 1). 1 1 1 1 1 a a a a a + + + +
  22. 22. Copyright © 2007 Pearson Education, Inc. Slide R-22 Simplifying Complex Fractions Solution 2 2 ( 1) ( 1) 11 1 11 1 1 1 1 11 1 1 1 ( 1) ( 1 1 ( 1) 1 ( 1) 2 ) 1 1 ( )( 1) a a a a a aa a a aa a a a a a a aa a a a a a a a a a a a a aa a   ++ + ÷ + + += =  + ++ ÷+ ++  + + + + + + + = + + + = + + +
  23. 23. Copyright © 2007 Pearson Education, Inc. Slide R-23 Now you try! • Pg 483 #’s 40

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