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Adding and Subtracting Rational Expressions with Unlike Denominators

Study Guide

Rational Expressions

A rational expression, also called an algebraic fraction, can be expressed as a quotient of polynomials. That is to say, $\frac{a}{b}$ where a and b are polynomials and b ≠ 0.

Important Notes

  • Just like fractions, we cannot directly add or subtract rational expressions with unlike denominators.
  • Find the Least Common Denominator (LCD) by factoring each denominator and taking the highest powers of each factor.
  • Rewrite each rational expression with the LCD as the new denominator.
  • Combine the numerators over the common denominator and simplify the result.
  • Example: To add $\frac{3}{x}$ and $\frac{2}{y}$: LCD = xy, rewrite as $\frac{3y}{xy} + \frac{2x}{xy}$, then combine to $\frac{3y + 2x}{xy}$.

Mathematical Notation

$a/b$ represents a fraction where a is the numerator and b is the denominator.$b \neq 0$ means that the denominator b cannot be equal to zero.Remember to use proper notation when solving problems

Why It Works

The process of adding and subtracting rational expressions is similar to that of fractions. The concept of finding a common denominator allows us to combine the expressions in a meaningful way.

Remember

Always simplify your final answer.

Quick Reference

Rational Expression:$\frac{a}{b}$ where a and b are polynomials and b ≠ 0

Understanding Adding and Subtracting Rational Expressions with Unlike Denominators

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

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Beginner Explanation

A rational expression is a fraction with polynomial expressions in the numerator and the denominator. Adding and subtracting them follows the same rules as adding and subtracting normal fractions.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

What is the sum of $\frac{3}{x}$ and $\frac{2}{y}$?

A$\frac{3y + 2x}{xy}$B$\frac{5}{x+y}$C$\frac{5}{xy}$DCannot be determinedCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Scenario Practice

Question ExerciseBeginner

Paint Mixing Scenario

Sam is mixing paint. He needs to combine a can of paint that is $\frac{2}{3}$ full with a can that is $\frac{1}{2}$ full.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

What is the result of subtracting $\frac{2}{x^2}$ from $\frac{3}{x}$? Hint: factor denominators if needed, find the LCD = x², rewrite each fraction, subtract, then simplify.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

What is the result of adding $\frac{3}{x^2-1}$ and $\frac{2}{x-1}$?

A$\frac{2x + 5}{x^2 - 1}$B$\frac{3 + 2(x + 1)}{x^2 - 1}$C$\frac{5}{x^2}$DCannot be determinedCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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