Rational Expressions Topic 3: Adding and Subtracting Rational Expressions.
MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and...
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Transcript of MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and...
![Page 1: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/1.jpg)
MTH095MTH095Intermediate AlgebraIntermediate Algebra
Chapter 7 – Rational Expressions
Section 7.3 – Addition and Subtraction of Rational Expressions
Copyright © 2010 by Ron Wallace, all rights reserved.
![Page 2: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/2.jpg)
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
![Page 3: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/3.jpg)
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
!
![Page 4: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/4.jpg)
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
![Page 5: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/5.jpg)
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.
![Page 6: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/6.jpg)
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
![Page 7: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/7.jpg)
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.
![Page 8: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/8.jpg)
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
![Page 9: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/9.jpg)
““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.
![Page 10: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/10.jpg)
““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.
![Page 11: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/11.jpg)
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!
![Page 12: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/12.jpg)
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
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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!
![Page 14: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/14.jpg)
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.
![Page 15: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/15.jpg)
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.
![Page 16: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/16.jpg)
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.
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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.
![Page 18: MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Section 7.3 – Addition and Subtraction of Rational Expressions Copyright © 2010 by Ron Wallace,](https://reader030.fdocuments.net/reader030/viewer/2022032723/56649d005503460f949d19ec/html5/thumbnails/18.jpg)
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.