Adding and Subtracting Rational Expressions with Different Denominators: Videos & Practice Problems

Adding and subtracting rational expressions requires finding the least common denominator (LCD), similar to working with rational numbers. The LCD is found by factoring denominators, often involving binomials like $x+4$ and $x-8$. Equivalent rational expressions are created by multiplying numerator and denominator by missing factors, enabling combination of terms. Simplifying involves distributing coefficients and combining like terms, resulting in a single rational expression. Understanding terms, coefficients, and polynomial degrees aids in mastering these operations, essential for manipulating multivariable polynomials and expressions in standard form.

Adding and Subtracting Rational Expressions with Unlike Denominators

Adding and subtracting rational expressions with unlike denominators involves a process similar to working with rational numbers. The key step is to find the least common denominator (LCD), which allows you to rewrite each rational expression as an equivalent expression with a common denominator. For example, when adding fractions like \(\frac{1}{30}\) and \(\frac{1}{20}\), the LCD is 60. Multiplying the numerator and denominator of each fraction by the missing factor (2 for \(\frac{1}{30}\) and 3 for \(\frac{1}{20}\)) converts them to \(\frac{2}{60}\) and \(\frac{3}{60}\), respectively. Adding these gives \(\frac{5}{60}\), which simplifies to \(\frac{1}{12}\).

This same principle applies to rational expressions. Suppose you have expressions with denominators \$30x\( and \)20x^2\(. The LCD here is \)60x^2\(. To rewrite each expression with this common denominator, multiply the numerator and denominator of \)\frac{1}{30x}\( by \)2x\( to get \)\frac{2x}{60x^2}\(, and multiply the numerator and denominator of \)\frac{1}{20x^2}\( by 3 to get \)\frac{3}{60x^2}\(. Adding these yields \)\frac{2x + 3}{60x^2}\(, which is the combined rational expression.

When working with polynomial denominators, factor each denominator completely to find the LCD. For instance, with denominators \)x + 4\( and \)x - 8\(, the LCD is their product \)(x + 4)(x - 8)\(. To add or subtract rational expressions like \)\frac{2}{x + 4}\( and \)\frac{7}{x - 8}\(, multiply the numerator and denominator of each by the missing factor: multiply \)\frac{2}{x + 4}\( by \)\frac{x - 8}{x - 8}\( and \)\frac{7}{x - 8}\( by \)\frac{x + 4}{x + 4}\(. This gives equivalent expressions with the common denominator \)(x + 4)(x - 8)$. Combining the numerators and simplifying involves distributing and combining like terms:

\[\frac{2(x - 8) - 7(x + 4)}{(x + 4)(x - 8)} = \frac{2x - 16 - 7x - 28}{(x + 4)(x - 8)} = \frac{-5x - 44}{(x + 4)(x - 8)}.\]

This final expression is the fully simplified result of the subtraction.

Understanding how to find the least common denominator, rewrite rational expressions as equivalent expressions, and then combine and simplify the numerators is essential for adding and subtracting rational expressions. Mastery of these steps enables you to handle more complex algebraic fractions confidently and lays the foundation for further work with rational functions.

Add the following expressions and simplify if possible:
$\frac{2}{x^2-1}+\frac{3}{x+1}$

• A. $\frac{3x+1}{\left(x+1\right)\left(x-1\right)}$
• B. $\frac{3x-1}{\left(x+1\right)\left(x-1\right)}$
• C. $\frac{3x-1}{x^2-1}$
• D. $\frac{3x-1}{x^2+1}$

Answer: (B)

Add the following expressions and simplify if possible:
$\frac{x}{x^2+3x}+\frac{3}{x+3}$

• A. $\frac{4}{x^2+3x}$
• B. $\frac{4x}{x+3}$
• C. $\frac{4}{x+3}$
• D. $\frac{4}{x}$

Answer: (C)

Add the following expressions and simplify if possible:
$\frac{x^2}{x^2-4}+\frac{2x}{4-x^2}$

• A. $\frac{x^2}{\left(x-2\right)\left(x+2\right)}$
• B. $\frac{x}{x-2}$
• C. $\frac{1}{x+2}$
• D. $\frac{x}{x+2}$

Answer: (D)

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{3}{x}-\frac{2}{x+2}$

• A. $\frac{3x+6}{x^2+2x}$
• B. $\frac{x+6}{x+2}$
• C. $\frac{x+6}{x^2+2x}$
• D. $\frac{3}{x^2+2x}$

Answer: (C)

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x+2}{x^2-4}-\frac{1}{x-2}$

• A. $\frac{1}{\left(x-2\right)\left(x+2\right)}$
• B. $\frac{x+2}{\left(x-2\right)\left(x+2\right)}$
• C. $\frac{x-2}{\left(x-2\right)\left(x+2\right)}$
• D. $0$

Answer: (D)

Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x^2}{x^2-25}-\frac{5x}{25-x^2}$

• A. $\frac{x}{x+5}$
• B. $\frac{x}{x-5}$
• C. $\frac{x}{\left(x-5\right)\left(x+5\right)}$
• D. $\frac{x+5}{x-5}$

Answer: (B)

Adding and Subtracting Rational Expressions with Unlike Denominators Example 1

When working with rational expressions, it is essential to understand how to add and subtract them by first factoring all components and then finding the least common denominator (LCD). The LCD is the product of all unique prime factors present in the denominators. For example, if the denominators include terms like x and x + 1, the LCD will be x(x + 1), ensuring all fractions share a common base for addition or subtraction.

To rewrite each rational expression with the LCD, multiply the numerator and denominator by any missing factors from the LCD. For instance, if a fraction has denominator x + 1 but the LCD is x(x + 1), multiply numerator and denominator by x. Once all expressions have the same denominator, combine the numerators accordingly. Simplify the resulting expression by expanding, combining like terms, and factoring where possible. For example, an expression like x² + x can be factored as x(x + 1), which may cancel with the denominator, simplifying the entire expression to 1.

Another common technique involves recognizing special factoring patterns such as the difference of squares. The difference of squares formula states that a² - b² = (a + b)(a - b). For example, x² - 4 can be factored as (x + 2)(x - 2). This factoring is crucial when determining the LCD and simplifying rational expressions involving quadratic terms.

After factoring, identify the LCD by including each unique factor only once. For denominators like x - 2 and x + 2, the LCD is their product (x + 2)(x - 2). Adjust each fraction to have this common denominator by multiplying numerator and denominator by the missing factor. Then, combine the numerators over the common denominator and simplify by distributing and combining like terms. Pay close attention to negative signs during distribution to avoid errors.

For example, when combining numerators such as (x - 1) + 2(x - 2) - 1(x + 2), distribute the constants to get x - 1 + 2x - 4 - x - 2. Combine like terms carefully: the constants sum to -7 and the x terms simplify to 2x, resulting in a numerator of 2x - 7. The final simplified expression is then (2x - 7) / ((x + 2)(x - 2)).

Mastering these steps—factoring, finding the LCD, rewriting expressions with the common denominator, combining numerators, and simplifying—enables efficient addition and subtraction of rational expressions. This process not only simplifies complex algebraic fractions but also reinforces understanding of polynomial factorization and arithmetic operations with rational expressions.

Here’s what students ask on this topic:

How do you find the least common denominator (LCD) when adding or subtracting rational expressions with different denominators?

To find the least common denominator (LCD) when adding or subtracting rational expressions, first factor each denominator completely. The LCD is the product of all unique factors from each denominator, including the highest powers of any repeated factors. For example, if the denominators are $x_2(x+3)$ and $x+3$, the LCD would be $x_2(x+3)$. After finding the LCD, rewrite each rational expression as an equivalent expression with the LCD as the denominator by multiplying numerator and denominator by the missing factors. This process ensures the denominators match, allowing you to add or subtract the numerators directly.

What steps should I follow to add rational expressions with unlike denominators?

To add rational expressions with unlike denominators, follow these steps: (1) Find the least common denominator (LCD) by factoring each denominator and combining all unique factors. (2) Rewrite each rational expression as an equivalent expression with the LCD as the denominator by multiplying numerator and denominator by the missing factors. (3) Once denominators are the same, add the numerators together while keeping the common denominator. (4) Simplify the resulting expression by distributing any coefficients and combining like terms in the numerator. (5) Finally, factor and reduce the expression if possible. This method is similar to adding fractions but involves polynomial expressions.

How do you simplify the numerator after adding or subtracting rational expressions?

After adding or subtracting rational expressions, simplify the numerator by first distributing any coefficients across terms inside parentheses. For example, if the numerator is $(2x-8)-(7x+4)$, distribute to get $2x-16-7x-28$. Then combine like terms, such as $2x-7x=-5x$ and constants $-16-28=-44$. The simplified numerator becomes $-5x-44$. This simplification makes the rational expression easier to work with and helps in further factoring or reducing.

Can you explain how to write rational expressions as equivalent expressions with a common denominator?

To write rational expressions as equivalent expressions with a common denominator, first identify the least common denominator (LCD) of the given expressions. Then, for each rational expression, determine the missing factors needed to reach the LCD. Multiply both the numerator and denominator of each expression by these missing factors. For example, if one expression has denominator $x+4$ and the LCD is $x+4\left)\right(x-8$, multiply numerator and denominator by $x-8$. This process creates equivalent expressions with the same denominator, allowing you to add or subtract the numerators directly.

What is the process to subtract rational expressions with different denominators?

Subtracting rational expressions with different denominators involves several steps: (1) Find the least common denominator (LCD) by factoring each denominator and combining all unique factors. (2) Rewrite each rational expression as an equivalent expression with the LCD by multiplying numerator and denominator by the missing factors. (3) Subtract the numerators while keeping the common denominator. (4) Simplify the numerator by distributing any negative signs and combining like terms. (5) Factor and reduce the resulting expression if possible. This method ensures the expressions have a common denominator, making subtraction straightforward and the final expression simplified.