Algebra II Chapter 8 Notes Name: ______
Notes 11: 8.4 Multiply and Divide Rational Expressions
A "rational expression" is a ______of two polynomials. A rational expression is in simplified form if its numerator and denominator have no common factors other than ______. Simplifying Rational Expressions (Fractions): Factor all expressions (numerators and denominators) ______(look for ______!) Reduce by canceling out ______(terms in parentheses) from the numerator and denominator 4 - x Look to cancel opposites; leave a (-1) behind. Ex: = x - 4 Re-write answer as a new and reduced fraction
Example 1 Simplify a rational expression
Factor numerator and denominator.
= Divide out common factor.
= Simplified form. ______
Practice
x2 - 4 2x2 + 2 x - 40 3x3+ 6 x 2 + 12 x 1. 2. 3. x2 +9 x + 14 x2 +2 x - 15 x3 -8
1 x2 -11 x + 24 3x2 - 108 x3-5 x 2 - 3 x + 15 4. 5. 6. x2 -3 x - 40 x2 +12 x + 36 x2 -8 x + 15
MULTIPLYING RATIONAL EXPRESSIONS Multiplying/Dividing Rational Expressions: Factor all expressions (numerators and denominators) ______(look for ______!) If a division problem, ______the second fraction and write as a ______problem. Multiply the ______together and multiply the ______together (just write factors next to each other; don’t actually FOIL) Reduce by canceling out ______(terms in parentheses) from the numerator and denominator [Look for ______and ways to ______!!!] Re-write answer as a new and reduced fraction
Example 2 Multiply rational expressions 2x2+ 4 x x 2 - 9 x + 18 x2 -4 x - 12g 2x
Factor numerator and = denominator.
= Multiply numerator and denominator.
= ______Divide out common factors and write in simplified form.
2 Example 3 Divide rational expressions 3 8x2 - 8 x x+7 x2 + 6 x - 7
Multiply by reciprocal.
Factor.
= Divide out common factors and ______write in simplified form.
Practice
5 3 2 2 4x+ 5 x x + 1 48x y x y x-3 x - 10 2 ( ) ( ) 7. 4g 3 2 8. g(x+ 10 x + 21) 9. 2 g y6 x y x2 -2 x - 15 x2( x + 5)
3 2 x -8 ( y - 4) ( x - 2) 2 10. g 11. g( x-9 x + 14) ( y - 2) ( x2 +2 x + 4) 2( x - 7)
3 2 2 2 8x y z 10 xy x-4 x - 5 2 12. 3 4 13. �( x+ 6 x 5) xz x z x + 5
2 2 x-8 x + 15 2 ( x- x - 20) 14. 2 �( x- x 20) 15. �( x 4) x+ 4 x x + 5
2 3 2 x-100 x - 10 x 2 x+ 5 x 2 16. g x 17. 4x2 x x+5 x + 4
4 Notes 12: 8.5 Add and Subtract Rational Expressions (non complex)
Add and Subtract Rational expressions Add (or subtract) with LIKE DENOMINATORS
When adding/subtracting fractions, you need a ______ Add/Subtract the numerators (if you are subtracting, be sure to ______the subtraction sign). Leave the denominators ______and ______ Factor and reduce, if possible
Example 1: Example 2:
7 2(___) ( ___) __ 5 3x - 4 (______) ___( _____) + = = = - = = 6x 6 x ( ___) ( ___) __ ( x+2) ( x + 2) ( _____) ( _____)
Practice:
x 4 3y2 5 x 17 12 1. - 2. + 3. - 16x2 16 x 2 x+5 x + 5 2x- 3 2 x - 3
2x 8 4 n 4. - 5. + x3-3 x 3 - 3 x-3 x - 3
5 FINDING THE LCD: To add (or subtract) two rational expressions with unlike denominators, find the ______(LCD), which is the ______(LCM) of the denominators.
Finding the Least Common Multiple
24x2 and 8x2 - 16 x
1. Factor each polynomial and write numerical factors as product of primes
2. Form the LCM by writing each factor to the highest power it occurs in either polynomial
Find the LCM
6. 16( x2 - 9) and ( x - 3) 7. ( x2 -7 x + 10) , ( x - 5) and ( x + 7)
8. 4x2 and 2x2 + 2 x 9. 8 , 24 and 7
6 Add with UNLIKE DENOMINATORS Example 3:
7 x + 9x2 3 x 2 + 3 x
Step 1. Find the LCM of the denominators.
Step 2. In this step we are really just multiplying by “1”. Multiply each denominator by the term needed to “turn it into” the LCM. Multiply the numerator by that same term as well.
Step 3. Multiply
Step 4. Add numerators
7 Subtract with UNLIKE DENOMINATORS Example 4
12 3 - x2 +5 x - 24 x + 3
Step 1. Find the LCM of the denominators
Step 2. In this step we are really just multiplying by “1”. Multiply each denominator by the term needed to “turn it into” the LCM. Multiply the numerator by that same term.
Step 3. Multiply
Step 4. Subtract numerators.
8 Practice
4 2 3x 6 2 10. + 11. - - x-3 x + 6 x2 + x -12 x + 4 x - 3
4 2x 3x 4 x 12. + 13. + x+5 x2 - 25 x+2 4( x + 2)
A ______is a fraction that contains a fraction in its numerator or denominator. A complex fraction can be simplified in 2 ways.
Method 1 – Write the numerator and denominator as a single fraction. Then divide the numerator by the denominator.
Method 2 – Multiply the numerator and denominator by the LCM of every fraction in the numerator and denominator. Then simplify.
1 Example 1 (simplify using method 1): 5 2 - 2x x
5 Example 2 (simplify using method 2): x + 4 1 2 + x+ 4 x
Notes 13: 8.5 Add and Subtract Rational Expressions (complex)
9 Practice
2 3 + 1. x x -1 1 2x - 2
3 2 + 2. x + 2 3 2x 1 - x+ 2 x
10 3x - 2 3. 2x - 1 5 x - 4x 2 x - 1
11 6x x 5 7 - + 2 4. 4 4 5. 2x x 8 3x 3 1 + - 2 2 x2 2 x
12 Notes 14: 8.6 Solve Rational Equations
You can use ______to solve a rational equation when each side of the equation is a single fraction.
When solving a rational equation, ______be sure to check for ______solutions by plugging your answer or answers back into the original equation.
9 4 Example 1: = 3x x + 2
Practice
1 x 3 4x 1. = 2. = 2x+ 5 11 x + 8 x-1 1 - x
13 Solving Rational Equations
Factor denominators, find ______ Set factored common denom = 0 and solve; these are the restrictions on the domain Multiply to make all denominators match (be sure to multiply on top and bottom!) Cancel out matching denominators; solve the left-over equation If you are left with a quadratic; solve by ______or ______ Compare your solutions to the restrictions to check for extraneous solutions (OR plug in your x values to make sure the equation “works”
3 4x + 1 Example 2: + = LCD = ______, x ______2x- 1 x - 1
5x 14 2( x + 7) 2x + 20 3. -2 = 4. -2 = x-1 x2 - 1 x+4 2 x + 8
14 10x + 9 5. +3 = x x - 4
3x 12 6. = + 2 x+1 x2 - 1
15 3 3 7.) + =1 x-4 x + 4
3 5 8.) = 2x x - 7
3 1 1 9.) - = x+1 x - 2 x2 - x - 2
1 1 4 10.) + = y-2 y + 2 y2 - 4
16 HW#15: Chapter 8 Review Sheet
Please complete each of the following problems on a separate sheet of paper.
For questions 1-3, simplify each rational expression. x2 -8 x + 12 x2 -2 x + 1 x2 - 25 1. 2. 3. x2 +3 x - 10 x2 -1 x3 -125
For questions 4-15, perform the indicated operation and simplify your result. 7x2- 21 x x 2 2x3- 12 x 2 x 2 + 9 x + 18 4. 5. x2 -2 x - 35 x - 7 x2-4 x - 12 8 x 3 + 24 x 2
x2-100 x 3 - 5 x 2 - 50 x ( x - 10) 2 x2+2 x - 35 3 x 2 - 12 6. 赘 7. 4x2 x 4+ 10 x 3 5x x2-7 x + 12 x 2 - 13 x + 40
3x + 1 5 x3 x 8. - 9. + x2 - x -12 3x - 12 x -1 x2 -1
x 1 - 3x- 8 5 x 2 x - 4 4 10. + + 11. x2+6 x + 9 x 2 - 9 x - 3 9 x2 + 4x x - 4
2 5 2 1 - + 3x- 1 12 x + 4 2 x - 2 12. 13. x - 4 x 5 3 + 9x2 - 1 x-2 x + 2
8x - 1 4 4- 9x 1 14. - 15. - x2 + x - 6 x - 2 x+5 2 x - 1
For questions 16-21, solve each equation and check your solutions. -2 1 5x x2 16. = 17. = x+3 x + 1 10-x x - 10
3 2 x 2x 4 x- 1 17 x + 4 18. = + 19. - = x2 - 4 x+2 x - 2 x-2 3 x + 2 3x2 - 4 x - 4
3x - 2 13 2x + 8 2 20. + = 21. - = x x+1 3 3 x + 3 x+3 x + 3
**Please check and correct your work using the online key**
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