Problem 63 - Algebra A Combined Function

Question

Choose the correct \(L C D\) of \(\frac{11 a^{3}}{4 a-20}\) and \(\frac{15 a^{3}}{(a-5)^{2}}\). a. \(4 a(a-5)(a+5)\) b. \(a-5\) c. \((a-5)^{2}\) d. \(4(a-5)^{2}\) e. \((4 a-20)(a-5)^{2}\)

Step-by-Step Solution

Answer

The correct LCD is (d) \(4(a - 5)^2\).

Step 1: Understand the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest expression that both denominators can divide into evenly. It includes all factors of each denominator, using the highest power of shared factors.

Step 2: Identify the Denominators

Identify the denominators of the given fractions. The denominators are: 1. For the first fraction: \(4a - 20\). 2. For the second fraction: \((a - 5)^2\).

Step 3: Factor the Denominators

Find the factors of each denominator. 1. The first denominator \(4a - 20\) factors to \(4(a - 5)\).2. The second denominator \((a - 5)^2\) is already in its factored form.

Step 4: Determine the LCD

To find the LCD, take the highest power of each factor present in both denominators:- The factor \((a - 5)\) appears in both and must be at least squared, as \((a - 5)^2\) in the second denominator has this factor twice.- The factor \(4\) is present in the first denominator.Thus, the LCD is \(4(a - 5)^2\).

Step 5: Verify the Correct Answer

Compare the determined LCD with the given options.The correct answer is option (d): \(4(a - 5)^2\) which matches our determined LCD.

Key Concepts

Factoring Expressions

Factoring expressions is a technique used to simplify algebraic expressions by expressing them as a product of simpler factors. This process is a crucial step when you want to find the least common denominator (LCD) of algebraic fractions. For example, consider the denominator in the fraction \(\frac{11 a^{3}}{4 a-20}\). To factor \(4a - 20\), you look for a common factor in each term. In this case, \(4\) is a common factor:

• Write each term: \(4a\) and \(20\)
• Factor out \(4\): \(4(a - 5)\)

By factoring, you make calculations involving the expression more manageable and reveal common factors that play a significant role in determining the LCD. Remember, the key is finding the greatest common factor, which simplifies the problem considerably.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Understanding how to handle these is essential in algebra, especially when combining fractions or finding a common denominator. Consider the equation \(\frac{11 a^{3}}{4 a-20}\) and \(\frac{15 a^{3}}{(a-5)^{2}}\) :

• Numerators like \(11a^3\) and \(15a^3\) consist of variables raised to a power.
• Denominators like \(4a - 20\) and \((a-5)^2\) are also algebraic expressions that can be factored.

The challenge with algebraic fractions is managing the algebraic part when performing operations like addition or finding the LCD. This involves factoring expressions and identifying common elements in the denominators.

Denominators

Denominators are the key to comparing and working with fractions, including algebraic ones. They inform us about the quantity into which the numerator is divided. When dealing with the task of finding the least common denominator (LCD), it is crucial to understand all parts of each denominator:

• For \(\frac{11a^3}{4a-20}\), the denominator can be rewritten as \(4(a - 5)\).
• For \(\frac{15a^3}{(a-5)^2}\), the denominator is already factored as \((a-5)^2\).

The LCD must include all integer and variable factors of each expression. In this case, the LCD is \(4(a-5)^2\), considering both the need to cover the highest power of \((a - 5)\) which appears as \((a - 5)^2\) and includes the factor \(4\) from \(4a - 20\). Correctly identifying and using these components permits not only simplification but also effective management of algebraic equations.