Rational Expressions and Excluded Values

Explore original denominator, factoring and cancellation, equivalent restricted rules.

Lesson 1 of 53 parts Practice Flashcards

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Original Denominator

Objective: Explain original denominator and use it to solve or interpret a problem.

A rational expression is a quotient of polynomials with denominator nonzero. Record forbidden inputs from the original denominator before any cancellation. Cancellation can hide a forbidden input.

Worked examples

Try itRational Expressions and Excluded Values — practice 1.1: For (x−4)/(x−5), what x-value is excluded?Show answer

Step 1: The original denominator is x−5. Step 2: Set it equal to zero: x−5=0. Step 3: Exclude x=5.

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Original Denominator

Objective: Explain original denominator and use it to solve or interpret a problem.

A rational expression is a quotient of polynomials with denominator nonzero. Record forbidden inputs from the original denominator before any cancellation. Cancellation can hide a forbidden input.

Worked examples

Try itRational Expressions and Excluded Values — practice 1.1: For (x−4)/(x−5), what x-value is excluded?Show answer

Step 1: The original denominator is x−5. Step 2: Set it equal to zero: x−5=0. Step 3: Exclude x=5.

Factoring and Cancellation

Objective: Explain factoring and cancellation and use it to solve or interpret a problem.

Cancel common factors only when they are nonzero; x−r in numerator and denominator cancels for x≠r but the original expression remains undefined at r. No, the original expression was undefined there.

Worked examples

Try itRational Expressions and Excluded Values — practice 2.1: Simplify (x−2)(x+8)/(x−2) for x≠2. Evaluate at x=0.Show answer

Step 1: Cancel the shared factor only because x≠2. Step 2: The remaining expression is x+8. Step 3: At x=0, its value is 8.

Equivalent Restricted Rules

Objective: Explain equivalent restricted rules and use it to solve or interpret a problem.

Two formulas may agree on all shared inputs yet have different domains. An apparent removable hole is a missing point, not a vertical asymptote. A cancelled denominator factor leaves its original input excluded.

Worked examples

Try itRational Expressions and Excluded Values — practice 3.1: Evaluate (3x+7)/(x+1) at x=11.Show answer

Step 1: Substitute x=11: numerator 40, denominator 12. Step 2: Form (40)/(12). Step 3: Simplify to 10/3.

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