Learn on PengienVision, Algebra 2Chapter 4: Rational Functions

Lesson 4: Adding and Subtracting Rational Expressions

In this Grade 11 enVision Algebra 2 lesson from Chapter 4, students learn how to add and subtract rational expressions with both like and unlike denominators, including finding the least common multiple (LCM) of polynomials to identify the least common denominator (LCD). The lesson covers adding numerators when denominators are the same, rewriting expressions with the LCD when denominators differ, and simplifying results by factoring and canceling common factors. Students also work with compound fractions as they build fluency with operations on rational expressions.

Section 1

Add Rational Expressions with a Common Denominator

Property

If $p$ , $q$ , and $r$ are polynomials where $r \neq 0$ , then

\frac{p}{r} + \frac{q}{r} = \frac{p+q}{r}

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator. Always check if the resulting fraction can be simplified.

Examples

Add: $\frac{5x}{x+2} + \frac{10}{x+2}$ .

This equals $\frac{5x+10}{x+2}$ . Factoring the numerator gives $\frac{5(x+2)}{x+2}$ , which simplifies to $5$ .

Add: $\frac{x^2}{x-5} + \frac{2x-35}{x-5}$ .

This equals $\frac{x^2+2x-35}{x-5}$ . Factoring the numerator gives $\frac{(x+7)(x-5)}{x-5}$ , which simplifies to $x+7$ .

Section 2

Find LCD of Polynomials Using Factorization

Property

The lowest common denominator (LCD) for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator.
To Find the LCD:

Factor each denominator completely.
Include each different factor in the LCD as many times as it occurs in any one of the given denominators.

Examples

Find the LCD for $\frac{5}{6x^2y}$ and $\frac{7}{9xy^3}$ . The factors are $2, 3^2, x^2, y^3$ . The LCD is $2 \cdot 3^2 \cdot x^2 \cdot y^3 = 18x^2y^3$ .
Find the LCD for $\frac{x}{x^2-16}$ and $\frac{3x}{x^2+8x+16}$ . The factored denominators are $(x-4)(x+4)$ and $(x+4)^2$ . The LCD is $(x-4)(x+4)^2$ .
Find the LCD for $\frac{1}{5a(a-2)^3}$ and $\frac{b}{10a^2(a-2)}$ . The factors are $2, 5, a^2, (a-2)^3$ . The LCD is $10a^2(a-2)^3$ .

Explanation

The LCD is the smallest shared 'target' denominator for unlike fractions. Find it by factoring all denominators and taking the highest power of each unique factor that appears. This ensures all original denominators divide into it evenly.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Add Rational Expressions with a Common Denominator

Property

If $p$ , $q$ , and $r$ are polynomials where $r \neq 0$ , then

\frac{p}{r} + \frac{q}{r} = \frac{p+q}{r}

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator. Always check if the resulting fraction can be simplified.

Examples

Add: $\frac{5x}{x+2} + \frac{10}{x+2}$ .

This equals $\frac{5x+10}{x+2}$ . Factoring the numerator gives $\frac{5(x+2)}{x+2}$ , which simplifies to $5$ .

Add: $\frac{x^2}{x-5} + \frac{2x-35}{x-5}$ .

This equals $\frac{x^2+2x-35}{x-5}$ . Factoring the numerator gives $\frac{(x+7)(x-5)}{x-5}$ , which simplifies to $x+7$ .

Section 2

Find LCD of Polynomials Using Factorization

Property

The lowest common denominator (LCD) for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator.
To Find the LCD:

Factor each denominator completely.
Include each different factor in the LCD as many times as it occurs in any one of the given denominators.

Examples

Find the LCD for $\frac{5}{6x^2y}$ and $\frac{7}{9xy^3}$ . The factors are $2, 3^2, x^2, y^3$ . The LCD is $2 \cdot 3^2 \cdot x^2 \cdot y^3 = 18x^2y^3$ .
Find the LCD for $\frac{x}{x^2-16}$ and $\frac{3x}{x^2+8x+16}$ . The factored denominators are $(x-4)(x+4)$ and $(x+4)^2$ . The LCD is $(x-4)(x+4)^2$ .
Find the LCD for $\frac{1}{5a(a-2)^3}$ and $\frac{b}{10a^2(a-2)}$ . The factors are $2, 5, a^2, (a-2)^3$ . The LCD is $10a^2(a-2)^3$ .

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Add Rational Expressions with a Common Denominator

Property

If $p$ , $q$ , and $r$ are polynomials where $r \neq 0$ , then

\frac{p}{r} + \frac{q}{r} = \frac{p+q}{r}

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator. Always check if the resulting fraction can be simplified.

Examples

Add: $\frac{5x}{x+2} + \frac{10}{x+2}$ .

This equals $\frac{5x+10}{x+2}$ . Factoring the numerator gives $\frac{5(x+2)}{x+2}$ , which simplifies to $5$ .

Add: $\frac{x^2}{x-5} + \frac{2x-35}{x-5}$ .

This equals $\frac{x^2+2x-35}{x-5}$ . Factoring the numerator gives $\frac{(x+7)(x-5)}{x-5}$ , which simplifies to $x+7$ .

Section 2

Find LCD of Polynomials Using Factorization

Property

The lowest common denominator (LCD) for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator.
To Find the LCD:

Factor each denominator completely.
Include each different factor in the LCD as many times as it occurs in any one of the given denominators.

Examples

Find the LCD for $\frac{5}{6x^2y}$ and $\frac{7}{9xy^3}$ . The factors are $2, 3^2, x^2, y^3$ . The LCD is $2 \cdot 3^2 \cdot x^2 \cdot y^3 = 18x^2y^3$ .
Find the LCD for $\frac{x}{x^2-16}$ and $\frac{3x}{x^2+8x+16}$ . The factored denominators are $(x-4)(x+4)$ and $(x+4)^2$ . The LCD is $(x-4)(x+4)^2$ .
Find the LCD for $\frac{1}{5a(a-2)^3}$ and $\frac{b}{10a^2(a-2)}$ . The factors are $2, 5, a^2, (a-2)^3$ . The LCD is $10a^2(a-2)^3$ .