Least Common Denominators: Videos & Practice Problems

Finding the least common denominator (LCD) for rational expressions involves factoring denominators into prime factors and variables, then selecting the highest powers of each unique factor. For example, the LCD of $\mathrm{30x}$ and $\mathrm{20x}{x}^2$ is $60x^2$. Writing equivalent expressions requires multiplying numerator and denominator by missing factors to match the LCD, ensuring common denominators for addition or subtraction of rational expressions. This process parallels working with rational numbers and polynomials.

Least Common Denominators of Rational Expressions

Adding or subtracting rational expressions often requires finding a common denominator, especially when the denominators are different. The key to this process is determining the Least Common Denominator (LCD), which is the smallest expression that both denominators can divide into without leaving a remainder. This concept extends from rational numbers to rational expressions, with the main difference being the inclusion of variables and polynomial factors.

To find the LCD, start by factoring each denominator completely into its prime factors and variable components. For example, the number 30 factors into 3 × 2 × 5, while 20 factors into 2² × 5. When variables are involved, such as in 30x or 20x², include the variable factors as well, like x or x². After factoring, identify all unique prime factors and variables, taking the highest power of each. For instance, between 2 and 2², the highest power is 2², and between x and x², the highest power is x².

The LCD is then the product of these unique factors raised to their highest powers. Using the previous example, the LCD of denominators 30x and 20x² is 3 × 2² × 5 × x², which simplifies to 60x². This method ensures the LCD is the smallest expression that both denominators can divide into, facilitating the addition or subtraction of rational expressions.

When dealing with polynomial denominators, such as quadratics, factor them into binomials or simpler polynomials. For example, the quadratic expression x² + 7x + 10 factors into (x + 2)(x + 5) because 2 and 5 multiply to 10 and add to 7. After factoring, list all unique factors, including binomials, and multiply them to find the LCD. If a factor appears in both denominators, include it only once, but with the highest exponent if applicable.

In summary, finding the least common denominator involves fully factoring each denominator, identifying unique factors with their highest powers, and multiplying these factors together. This process is essential for simplifying the addition or subtraction of rational expressions and builds on the foundational skills of factoring and prime factorization. Mastery of this technique enables efficient manipulation of algebraic fractions and prepares you for more complex algebraic operations.

Find the least common denominators of the rational expressions:
$\frac{2y^2+3y-5}{y^2-4}$, $\frac{y-2}{y^2+y-6}$

• A. $\left(y-2\right)\left(y-2\right)\left(y+3\right)$
• B. $y^2-4$
• C. $y^2+y-6$
• D. $\left(y+2\right)\left(y-2\right)\left(y+3\right)$

Answer: (D)

Find the least common denominators of the rational expressions:
$\frac{3x^2-12}{x^2-x-6}$, $\frac{5x+10}{x^2-4x+3}$

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

Answer: (C)

Find the least common denominators of the rational expressions:
$\frac{2x^2+5x-3}{x^2-9},\frac{4x-12}{x^2-x-6},\frac{x^2+x-2}{x^2-4x+3}$

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

Answer: (A)

Write Equivelant Expressions With Common Denominators

Understanding how to write equivalent rational expressions with common denominators builds directly on the concept of finding the least common denominator (LCD). The LCD is essential because it allows us to rewrite rational expressions so they share the same denominator, enabling addition, subtraction, or comparison. This process mirrors the method used with rational numbers, where the LCD is found by breaking down denominators into their prime factors and combining the highest powers of these primes.

For example, when working with rational numbers such as 1/30 and 1/20, the LCD is determined by factoring each denominator into primes: 30 factors into 2 × 3 × 5, and 20 factors into 2² × 5. The LCD must include the highest powers of each prime, resulting in 2² × 3 × 5 = 60. To rewrite each fraction with this common denominator, identify the missing prime factors in each denominator compared to the LCD. For 30, the missing factor is 2 (since 30 has one 2 but the LCD has 2²), and for 20, the missing factor is 3. Multiplying numerator and denominator of 1/30 by 2/2 gives 2/60, and multiplying 1/20 by 3/3 gives 3/60, creating equivalent fractions with a common denominator of 60.

This same principle applies to rational expressions involving variables. Consider denominators like 30x and 20x². Factoring these, 30x is 2 × 3 × 5 × x, and 20x² is 2² × 5 × x². The LCD must include the highest powers of each prime and variable factor, which is 2² × 3 × 5 × x² = 60x². To find missing factors, compare each denominator to the LCD: 30x is missing a 2 and an x (since the LCD has 2² and x²), while 20x² is missing a 3. Multiplying 1/(30x) by (2x)/(2x) yields (2x)/(60x²), and multiplying 1/(20x²) by 3/3 yields 3/(60x²). Both expressions now share the common denominator 60x², making them equivalent rational expressions.

This method emphasizes prime factorization and careful comparison of denominators to the LCD, ensuring all missing factors are accounted for. By multiplying numerator and denominator by these missing factors, rational expressions are rewritten with a common denominator, facilitating further operations. Mastery of this process is crucial for simplifying, adding, or subtracting rational expressions effectively.

Rewrite the expression $\frac{3}{x+2}$ into an equivalent expression having a denominator of $(x+2)(x+5)$.

• A. $\frac{3x+5}{\left(x+2\right)\left(x+5\right)}$
• B. $\frac{3x^2+15}{\left(x+2\right)\left(x+5\right)}$
• C. $\frac{3x+15}{\left(x+2\right)\left(x+5\right)}$
• D. $\frac{3x^2+5}{\left(x+2\right)\left(x+5\right)}$

Answer: (C)

Rewrite the expression $\frac{7x^2+7x}{x^2-1}$ into an equivalent expression having a denominator of $x-1$.

• A. $\frac{7x^2}{x-1}$
• B. $\frac{7x^2+7}{x-1}$
• C. $\frac{7x}{x-1}$
• D. $\frac{7x^2-7}{x-1}$

Answer: (C)

Rewrite the expression $\frac{2x^2+2x}{-x^2+1}$ into an equivalent expression having a denominator of $x-1$.

• A. $\frac{2x}{x-1}$
• B. $-\frac{2x}{x-1}$
• C. $\frac{2x^2}{x-1}$
• D. $-\frac{2x^2}{x-1}$

Answer: (B)

Write Equivelant Expressions With Common Denominators Example 1

To find the least common denominator (LCD) and rewrite rational expressions with this common denominator, start by identifying all unique prime factors present in each denominator. For example, if the denominators include factors like 3, 7, x, x², and y, list each unique factor, paying special attention to the highest powers of variables. In this case, since both x and x² appear, use the higher power, x², in the LCD.

Once the unique prime factors are identified, multiply them together to find the LCD. For instance, multiplying 3, 7, x², and y results in the LCD: \$21x^{2}y\(.

Next, convert each rational expression to an equivalent form with the denominator equal to the LCD. To do this, determine the missing factors in each denominator compared to the LCD. Multiply both the numerator and denominator of each expression by these missing factors to maintain equivalence.

For example, if one expression has a denominator of \)3x\(, it is missing factors of 7, x, and y to reach the LCD \)21x^{2}y\(. Multiply numerator and denominator by \)7xy\(:

\[\frac{2}{3x} \times \frac{7xy}{7xy} = \frac{14xy}{21x^{2}y}\]

For another expression with denominator \)7x^{2}y\(, the missing factor to reach the LCD is 3. Multiply numerator and denominator by 3:

\[\frac{5}{7x^{2}y} \times \frac{3}{3} = \frac{15}{21x^{2}y}\]

After these steps, both rational expressions share the common denominator \)21x^{2}y$, allowing for straightforward addition, subtraction, or comparison. This method ensures that rational expressions are rewritten with the least common denominator, simplifying further operations.

Write Equivelant Expressions With Common Denominators Example 2

When rewriting rational expressions to have a specific denominator, the key is to identify the missing factors needed to match the desired denominator, which acts as the least common denominator (LCD). For example, consider the expression \(\frac{9n}{13}\), and the goal is to rewrite it with the denominator \$13(n - 1)\(. Since the original denominator is 13, the missing factor is \)(n - 1)\(. To maintain equivalence, multiply both the numerator and denominator by \)(n - 1)\(, resulting in:

\[\frac{9n \times (n - 1)}{13 \times (n - 1)} = \frac{9n^2 - 9n}{13n - 13}\]

This process ensures the expression is equivalent but now has the denominator \)13(n - 1)\(.

In another example, consider the rational expression \)\frac{10k}{k + 4}\(, and the goal is to rewrite it with the denominator \)k^2 + 9k + 20\(. First, factor the quadratic denominator:

\[k^2 + 9k + 20 = (k + 4)(k + 5)\]

Since the original denominator is \)(k + 4)\(, the missing factor is \)(k + 5)\(. Multiply both numerator and denominator by \)(k + 5)$ to get:

\[\frac{10k \times (k + 5)}{(k + 4)(k + 5)} = \frac{10k^2 + 50k}{k^2 + 9k + 20}\]

This rewritten expression has the desired denominator and remains equivalent to the original. Recognizing how to factor polynomials and identify missing factors is essential for rewriting rational expressions with new denominators. This method leverages the concept of the least common denominator and ensures expressions are equivalent by multiplying numerator and denominator by the same factor, preserving the value of the rational expression.

Here’s what students ask on this topic:

What is the process for finding the least common denominator (LCD) of rational expressions?

Finding the least common denominator (LCD) of rational expressions involves a few key steps. First, factor each denominator completely into its prime factors and variables with their exponents. For example, if you have denominators like ${\mathrm{30}}_{}x$ and ${20}_{}{x}^2$, factor 30 into $2\; imes\; 3\; imes\; 5$ and 20 into $2^2\; imes\; 5$, then include the variables. Next, list all unique prime factors and variables, taking the highest power of each. For the example, the LCD would be $2^2\; imes\; 3\; imes\; 5\; imes\; x^2\; =\; 60x^2$. This LCD is the smallest expression that all denominators divide into, allowing you to add or subtract rational expressions with different denominators.

How do you write equivalent rational expressions with a common denominator?

To write equivalent rational expressions with a common denominator, start by finding the least common denominator (LCD) of the given expressions. Then, identify the missing factors in each denominator compared to the LCD. For each rational expression, multiply both the numerator and denominator by these missing factors. For example, if the LCD is $60x^2$ and one denominator is $30x$, the missing factors are $2x$. Multiply numerator and denominator by $2x$ to get an equivalent expression with denominator $60x^2$. Repeat this for all expressions. This process ensures all rational expressions have the same denominator, making addition or subtraction possible.

Why is factoring denominators important when finding the least common denominator?

Factoring denominators is crucial because it breaks down complex expressions into their prime factors and variables, revealing the building blocks of each denominator. This allows you to identify all unique factors and their highest powers across denominators. Without factoring, you might miss factors or powers needed for the least common denominator (LCD). For example, factoring $20x^2$ into $2^2\; imes\; 5\; imes\; x^2$ shows the powers of 2 and x explicitly. This ensures the LCD includes the highest powers of each factor, which is essential for correctly adding or subtracting rational expressions.

How do variables and their exponents affect the least common denominator?

Variables and their exponents play a key role in determining the least common denominator (LCD) because the LCD must include each variable raised to the highest exponent found in any denominator. For example, if one denominator has $x$ and another has $x^2$, the LCD must include $x^2$ to accommodate both. Ignoring the highest exponent would result in an incorrect LCD, making it impossible to properly add or subtract the rational expressions. So, when factoring denominators, always note the variables and their exponents and select the highest power for the LCD.

Can the process of finding the LCD for rational expressions be compared to finding the LCD for rational numbers?

Yes, the process is very similar. For rational numbers, you factor the denominators into prime numbers, list unique factors, and take the highest powers to find the LCD. For rational expressions, you do the same but also include variables and their exponents. The key difference is handling variables, but the principle remains: factor completely, identify unique factors with highest powers, and multiply them to get the LCD. This similarity helps in understanding and applying the concept across different types of expressions.