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Rational Expressions andEquations; Ratio and Proportion 6

Copyright © Cengage Learning. All rights reserved.

Section 6.36.3

Adding and Subtracting Rational Expressions

3

Objectives

Add two rational expressions with like denominators and write the answer in simplest form.

Subtract two rational expressions with like denominators and write the answer in simplest form.

Find the least common denominator (LCD) oftwo or more polynomials and use it to write equivalent rational expressions.

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22

33

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Objectives

Add two rational expressions with unlike denominators and write the answer in simplest form. Subtract two rational expressions with unlike denominators and write the answer in simplest form.

44

55

5

Add two rational expressions with like denominators and write the answer in simplest form

1.

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Add two rational expressions with like denominators and write the answer in simplest form

To add rational expressions with a common denominator, we follow the same process we use to add arithmetic fractions; add their numerators and keep the common denominator.

For example,

Add the numerators and keepthe common denominator.

2x + 3x = 5x

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Add two rational expressions with like denominators and write the answer in simplest form

In general, we have the following result.

Adding Rational Expressions with Like Denominators

If a, b, and d represent polynomials, then

provided the denominator

is not equal to 0.

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Example

Solution:

In each part, we will add the numerators and keep the common denominator.

a.

Add the numerators and keepthe common denominator.

Combine like terms.

because

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Example 1 – Solution

Add the numerators and keepthe common denominator.

Combine like terms.

cont’d

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Subtract two rational expressions with like denominators and write the answer in simplest form

2.

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Subtract two rational expressions with like denominators and write the answer in simplest form

To subtract rational expressions with a common denominator, we subtract their numerators and keep the common denominator.

Subtracting Rational Expressions with LikeDenominators

If a, b, and d represent polynomials, then

provided the denominator is not equal to 0.

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Example

Subtract, assuming no divisions by zero.

a.

b.

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Example – Solution

In each part, the rational expressions have the same denominator. To subtract them, we subtract their numerators and keep the common denominator.

a. Subtract the numerators and keepthe common denominator.

Combine like terms.

Divide out the common factor.

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Example – SolutionSubtract the numerators andkeep the common denominator.

Use the distributive property to remove parentheses.

Combine like terms.

cont’d

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Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions

3.

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Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions

Since the denominators of the fractions in the addition

are different, we cannot add the fractions in their present

form.

four-sevenths + three-fifths

To add these fractions, we need to find a common denominator.

The smallest common denominator (called the least or lowest common denominator) is the easiest one to use.

Different denominators

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Least Common Denominator

The least common denominator (LCD) for a set of fractions is the smallest number that each denominator will divide exactly.

We now review the method of writing two fractions using the LCD.

In the addition , the denominators are 7 and 5.

The smallest number that 7 and 5 will divide exactly is 35.

This is the LCD.

Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions

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We now build each fraction into a fraction with a denominator of 35.

Now that the fractions have a common denominator,we can add them.

Multiply numerator and denominator ofby 5, and multiply numerator anddenominator of by 7.

Do the multiplications.

Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions

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Example

Write each rational expression as an equivalent expression

with a denominator of 30y (y 0).

a.

b.

c.

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Example – Solution

To build each rational expression into an expression with a denominator of 30y, we multiply the numerator and denominator by what it takes to make the denominator 30y y (0).

a.

b.

c.

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There is a process that we can use to find the least common denominator of several rational expressions.

Finding the Least Common Denominator (LCD)

1. List the different denominators that appear in the rational expressions.

2. Completely factor each denominator.

3. Form a product using each different factor obtained in Step 2. Use each different factor the greatest number of times it appears in any one factorization. The product formed by multiplying these factors is the LCD.

Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions

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Add two rational expressions with unlike denominators and write the answer in simplest form

4.

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Add two rational expressions with unlike denominators and write the answer in simplest form

The process for adding and subtracting rational expressions with different denominators is the same as the process for adding and subtracting expressions with different numerical denominators.

For example, to add and , we first find the LCD of 7 and 5, which is 35.

We then build the rational expressions so that each one has a denominator of 35.

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Add two rational expressions with unlike denominators and write the answer in simplest form

Finally, we add the results.

Do the multiplications

Multiply numerator and denominator ofby 5, and numerator and denominator of by 7.

Add the numerators and keepthe common denominator.

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Add two rational expressions with unlike denominators and write the answer in simplest form

The following steps summarize how to add rational expressions that have unlike denominators.

Adding Rational Expressions with Unlike Denominators

To add rational expressions with unlike denominators:

1. Find the LCD.

2. Write each rational expression as an equivalent expression with a denominator that is the LCD.

3. Add the resulting fractions.

4. Simplify the result, if possible.

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Example

Add: , , and (b 0).

Solution:

The LCD of these rational expressions is

2 2 2 3 3 b = 72b.

To add the rational expressions, we first factor each

denominator:

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Example – Solution

In each resulting expression, we multiply the numerator and the denominator by whatever it takes to build the denominator to the lowest common denominator of

2 2 2 3 3 b.

Do the multiplications.

Add the fractions.

Simplify.

cont’d

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Subtract two rational expressions with unlike denominators and write the answer

in simplest form

5.

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Subtract two rational expressions with unlike denominators and write the answer in simplest form

To subtract rational expressions with unlike denominators,

we first write them as expressions with the same

denominator and then subtract the numerators.

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Example

Subtract: (x ≠ 0, –1)

Solution:

Because x and x + 1 represent different values and have

no common factors, the least common denominator (LCD)

is their product, (x + 1)x.

Build the fractions to obtainthe common denominator.

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Example – Solution

Subtract the numerators and keepthe common denominator.

Do the multiplication in the numerator.

cont’d