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}\(, and multiply \)\frac{1}{20x^2}\( by \)\frac{3}{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)\(. Combining the numerators involves distributing and simplifying: \)2(x - 8) - 7(x + 4) = 2x - 16 - 7x - 28 = -5x - 44\(. The final simplified expression is \)\frac{-5x - 44}{(x + 4)(x - 8)}$.

Understanding how to find the least common denominator and rewrite rational expressions as equivalent forms is essential for performing addition and subtraction with unlike denominators. This process ensures that the expressions share a common base, allowing the numerators to be combined directly. Simplifying the resulting expression by combining like terms and factoring when possible leads to a fully simplified rational expression. Mastery of these techniques is fundamental for working confidently with rational expressions in algebra.

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. For instance, if a fraction has denominator x + 1 but the LCD is x(x + 1), multiply numerator and denominator by x. After rewriting all fractions with the LCD, combine the numerators over the common denominator. Simplify the resulting expression by expanding, combining like terms, and factoring where possible. Often, terms in the numerator and denominator will cancel, simplifying the expression to its lowest terms.

In cases involving quadratic expressions, recognizing special factoring patterns such as the difference of squares is crucial. The difference of squares formula states that a² - b² = (a + b)(a - b). For example, x² - 4 factors into (x + 2)(x - 2). Factoring denominators into prime factors allows for accurate determination of the LCD, which is the product of all unique factors.

Once the LCD is established, adjust each rational expression by multiplying numerator and denominator by the missing factors to match the LCD. Then, combine the numerators over the common denominator. Carefully distribute multiplication over addition or subtraction in the numerators, paying close attention to negative signs to avoid errors. After combining like terms, simplify the numerator and write the final expression as a single rational expression.

For example, if the combined numerator simplifies to a linear expression like 2x - 7 and the denominator is the product of binomials such as (x + 2)(x - 2), the expression is simplified and ready for further use or evaluation.

Mastering the addition and subtraction of rational expressions involves a systematic approach: factor all expressions, find the least common denominator by identifying unique prime factors, rewrite each fraction with the LCD, combine numerators carefully, and simplify the resulting expression. This process ensures accurate and simplified results when working with rational expressions in algebra.

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, using the highest power of each factor present. For example, if the denominators are $x+4$ and $x-8$, the LCD is their product: $(x+4)(x-8)$. This LCD allows you to rewrite each rational expression with a common denominator, enabling addition or subtraction of the numerators. Remember, finding the LCD is similar to working with rational numbers but involves factoring polynomials.

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

To add rational expressions with unlike denominators, follow these steps: First, find the least common denominator (LCD) by factoring each denominator and combining all unique factors. Second, rewrite each rational expression as an equivalent expression with the LCD as the new denominator by multiplying the numerator and denominator by the missing factors. Third, add the numerators while keeping the common denominator. Finally, simplify the resulting expression by combining like terms and factoring if possible. For example, adding $\frac{2}{x}$ and $\frac{7}{x}$ involves rewriting each with the LCD $(x+4)(x-8)$, then adding the numerators and simplifying.

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

After adding or subtracting rational expressions, simplify the numerator by distributing any coefficients across parentheses and combining like terms. For example, if the numerator is $2(x-8)-7(x+4)$, distribute to get $\mathrm{2x}-16-\mathrm{7x}-28$. Then combine like terms: $(2x\; -\; 7x)\; +\; (-16\; -\; 28)\; =\; -5x\; -\; 44$. This simplified numerator is then placed over the common denominator to complete the expression.

Why is it important to rewrite rational expressions as equivalent expressions before adding or subtracting?

Rewriting rational expressions as equivalent expressions with a common denominator is essential because addition and subtraction require the denominators to be the same. Without a common denominator, the expressions cannot be directly combined. By multiplying the numerator and denominator of each rational expression by the missing factors of the LCD, you create equivalent expressions that share the same denominator. This process ensures the expressions are compatible for addition or subtraction, just like with regular fractions. It also helps maintain the equality of the expressions while enabling the combination of numerators.

Can you explain how to add rational expressions when the denominators are binomials?

When adding rational expressions with binomial denominators, first factor each denominator if possible. Then find the least common denominator (LCD) by multiplying the unique binomial factors. For example, if denominators are $x+4$ and $x-8$, the LCD is $(x+4)(x-8)$. Next, multiply the numerator and denominator of each expression by the missing binomial factor to create equivalent expressions with the LCD. Finally, add the numerators, simplify by distributing and combining like terms, and write the result over the LCD.