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.