Upload
eunice-flynn
View
218
Download
1
Tags:
Embed Size (px)
Citation preview
MTH095MTH095Intermediate AlgebraIntermediate Algebra
Chapter 7 – Rational Expressions
Section 7.3 – Addition and Subtraction of Rational Expressions
Copyright © 2010 by Ron Wallace, all rights reserved.
Review …Review …Adding Fractions
w/ Common (i.e. same) Denominators
Why?It’s just the distributive property!
a c a c
b b b
1 1 1 + a c a c
a c a cb b b b b b
3 8 1 1 1 3 8 11 + 3 8 3 8
5 5 5 5 5 5 5
Review …Review …Adding Fractions
w/ Common (i.e. same) Denominators
Subtracting Fractionsw/ Common (i.e. same) Denominators
a c a c
b b b
a c a c
b b b
Don’t forget
to simplify
!
Review …Review …Adding Fractions
wo/ Common (i.e. different) Denominators
Subtracting Fractionswo/ Common (i.e. different) Denominators
a c ad bc ad bc
b d bd bd bd
Don’t forget
to simplify
!
a c ad bc ad bc
b d bd bd bd
Least Common Multiple Least Common Multiple (LCM)(LCM) AKA: Least Common AKA: Least Common Denominator (LCD)Denominator (LCD)LCM = Smallest expression that
two other expressions divide into evenly.
With numbers …1. Factor w/ Primes2. LCM = product of each factor
raised to the highest power found in the factorization of the two numbers.
Least Common Multiple Least Common Multiple (LCM)(LCM) AKA: Least Common AKA: Least Common Denominator (LCD)Denominator (LCD) Examples with Numbers:
◦ Find the LCM of 20 & 70
◦ Find the LCM of 90 & 220
Least Common Multiple Least Common Multiple (LCM)(LCM) AKA: Least Common AKA: Least Common Denominator (LCD)Denominator (LCD)LCM = Smallest expression that
two other expressions divide into evenly.
With polynomials…1. Factor2. LCM = product of each factor
raised to the highest power found in the factorization of the two polynomials.
Least Common Multiple Least Common Multiple (LCM)(LCM) AKA: Least Common AKA: Least Common Denominator (LCD)Denominator (LCD) Examples with Polynomials:
◦ Find the LCM of x2 – 9 & 4x – 12
◦ Find the LCM of x3 + 2x2 – 3x & x4 – x2
““Un-Reducing” a FractionUn-Reducing” a FractionChange the following fraction into an
equivalent fraction with a denominator of 30.a. Factor both old & new denominators.b. Divide the new denominator by the old
denominator (i.e. cancel out factors).c. Multiply the old numerator by the result
of the above division.3
5 30
Our book calls this “building up” a fraction.
““Un-Reducing” a Rational Un-Reducing” a Rational ExpressionExpression
Change the following rational expression into an equivalent rational expression with a denominator of x(x+2)(x–2)2(x+3).
a. Factor both old & new denominators.b. Divide the new denominator by the old
denominator (i.e. cancel out factors).c. Multiply the old numerator by the result
of the above division.
2 2
3
4 ( 2)( 2) ( 3)
x
x x x x x
Our book calls this “building up” a fraction.
Adding Rational ExpressionsAdding Rational Expressionsw/ Common w/ Common (i.e. same)(i.e. same) DenominatorsDenominators
That is …1. Add numerators together.2. Keep the same denominator.3. Simplify (factor & cancel common
factors)
( ) ( ) ( ) ( )
( ) ( ) ( )
p x r x p x r x
q x q x q x
NOTE: Subtraction is the same, except that you subtract instead of add!
Adding Rational ExpressionsAdding Rational Expressionsw/ Common w/ Common (i.e. same)(i.e. same) DenominatorsDenominators
----- Examples -----
3 4 7 2
4 4
x x
2
2 2
2 15
9 9
x x
x x
Adding Rational ExpressionsAdding Rational Expressionswo/ Common wo/ Common (i.e. different)(i.e. different) DenominatorsDenominatorsThe Process (just like with numbers)…
1. Find the common denominator (LCD). This will be the denominator of the sum.
2. Un-Reduce both rational expressions so they end up with the same denominators (i.e. the LCD).
3. Add the fractions (they now have common denominators).
4. Simplify (factor & cancel common factors)
NOTE: Subtraction is the same, except that you subtract instead of add!
Adding Rational ExpressionsAdding Rational Expressionswo/ Common wo/ Common (i.e. different)(i.e. different) DenominatorsDenominators----- Examples -----
2 4 3
7 11
4 12xy x y z
1 of 5
The Process…1. Find the common denominator.2. Un-Reduce both rational expressions.3. Add the fractions.4. Simplify.
Adding Rational ExpressionsAdding Rational Expressionswo/ Common wo/ Common (i.e. different)(i.e. different) DenominatorsDenominators----- Examples -----
3 5
2
x
x x
2 of 5
The Process…1. Find the common denominator.2. Un-Reduce both rational expressions.3. Add the fractions.4. Simplify.
Adding Rational ExpressionsAdding Rational Expressionswo/ Common wo/ Common (i.e. different)(i.e. different) DenominatorsDenominators----- Examples -----
1 4
6 3 2
x x
x x
3 of 5
The Process…1. Find the common denominator.2. Un-Reduce both rational expressions.3. Add the fractions.4. Simplify.
Adding Rational ExpressionsAdding Rational Expressionswo/ Common wo/ Common (i.e. different)(i.e. different) DenominatorsDenominators----- Examples -----
2 25
5 5
x
x x
4 of 5
The Process…1. Find the common denominator.2. Un-Reduce both rational expressions.3. Add the fractions.4. Simplify.
Adding Rational ExpressionsAdding Rational Expressionswo/ Common wo/ Common (i.e. different)(i.e. different) DenominatorsDenominators----- Examples -----
2
3 7 11
2 1 2 7 3 3
x x
x x x x
5 of 5
The Process…1. Find the common denominator.2. Un-Reduce both rational expressions.3. Add the fractions.4. Simplify.