Perform Rational Expression Operations

Lesson 7 of 7: Arithmetic with Polynomials (+)

In this lesson:

• Multiply, divide, add, and subtract rational expressions
• Use the rational number analogy to justify each operation
• State and maintain domain restrictions throughout

Learning Objectives for This Lesson

By the end, you should be able to:

1. Multiply and divide, canceling common factors
2. Add and subtract with the same denominator
3. Add and subtract with different denominators using the LCD
4. State restrictions: exclude where denominators are zero
5. Explain operations via the rational number analogy

Recall: The Four Fraction Arithmetic Rules

• Multiply: — multiply tops, multiply bottoms
• Divide: — flip and multiply
• Add/Subtract: find LCD, build equivalent fractions, combine numerators

What is ? What LCD do you need?

Fraction Rules Already in Your Toolkit

— multiply tops, multiply bottoms

— flip the divisor

— common denominator

Every one of these rules works the same way for polynomial fractions.

From Long Division to Full Arithmetic

APR.D.6 converted a single rational expression to form.

Today: treat rational expressions as objects and perform all four operations on them.

Same objects, new operations: , , , with .

The Rational Number Analogy: Fraction Rules Apply

Learn the structure once — apply it to both integers and polynomials.

Domain Restrictions: Exclude Division by Zero

A rational expression is undefined wherever .

State restrictions before simplifying.

Example: — factor: ; domain: ,

After simplification, the restrictions remain even if the factor canceled.

Check-In: Find Domain of Each Expression

State the domain for each:

Factor denominators, set each factor equal to zero, state restrictions.

Multiply and Divide: Factor First, Then Cancel

Multiply: factor all numerators and denominators; cancel common factors; multiply.

Divide: rewrite as multiplication by the reciprocal; then factor and cancel.

Rule: cancel factors of the entire numerator/denominator — never cancel additive terms.

Worked: Factor First, Then Multiply

Factor: ,

Cancel and : ,

Worked: Flip and Multiply to Divide

Rewrite: . Factor: ;

Cancel , , : ,

Probe: Can You Cancel the ?

Is ?

Test at :

No. The in is a term, not a factor — it cannot be canceled.

Only factors of the entire numerator or denominator can cancel.

Why Addition Needs a Common Denominator

— denominators are different; you must find the LCD first.

Same principle for rational expressions:

Find the LCD: . Build equivalent fractions. Then add.

Building the LCD from Factored Denominators

How to find the LCD: factor denominators; take each distinct factor to its highest power.

• and : both factored → LCD
• and : → LCD

Worked: Add Fractions with Same Denominator

Add numerators, keep the denominator:

Factor: . Domain: .

Worked: Find LCD and Add Unlike Fractions

LCD . Build equivalent fractions:

Worked: Subtract with a Quadratic Denominator

Factor: . LCD .

Probe: Domain Restriction After Cancellation

Canceling gives — but what happens at ?

The original is undefined at . The simplified form is not. The domain restriction must be stated.

Check-In: Subtract and State Domain Restrictions

Compute . State domain restrictions.

Find the LCD, build equivalent fractions, subtract numerators, state restrictions.

Complex Problems: All Four Skills Together

Factor; simplify first term: . LCD .

Three Critical Errors to Prevent

Canceling terms: — is a term, not a factor

Dropping restrictions: only for

Sign error in subtraction: means subtract all of — put it in parentheses

Summary: Rational Expression Arithmetic Rules

Four operations, same rules as fractions:

• Multiply: factor, cancel, multiply remaining factors
• Divide: flip second fraction; factor, cancel, multiply
• Add/subtract (same denominator): combine numerators
• Add/subtract (different denominators): factor; find LCD; build equivalents; combine

State domain restrictions from the original; they survive all operations.

Next: Rational Equations and Partial Fractions

Rational equations (REI.A.2): set a rational expression equal to zero or another; clear denominators; solve; check for extraneous solutions.

Partial fractions (calculus): the reverse of addition — decompose back into .

Everything in this lesson feeds both directions.