Problem 16 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{3}{a^{2}+4 a}+\frac{5}{a^{2}-4 a} $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{8(a+1)}{a(a+4)(a-4)} \).

Step 1: Identify the Denominators

The given problem is \( \frac{3}{a^2 + 4a} + \frac{5}{a^2 - 4a} \). The denominators here are \( a^2 + 4a \) and \( a^2 - 4a \). These can be factored as: \( a^2 + 4a = a(a + 4) \) and \( a^2 - 4a = a(a - 4) \).

Step 2: Find the Least Common Denominator (LCD)

To add the fractions, we need a common denominator. The least common denominator of \( a(a+4) \) and \( a(a-4) \) is \( a(a+4)(a-4) \).

Step 3: Rewrite the Fractions with the LCD

We rewrite each fraction over the common denominator: \( \frac{3}{a(a+4)} = \frac{3(a-4)}{a(a+4)(a-4)} \) and \( \frac{5}{a(a-4)} = \frac{5(a+4)}{a(a+4)(a-4)} \).

Step 4: Simplify and Add the Numerators

Combine the numerators: \( 3(a-4) + 5(a+4) = 3a - 12 + 5a + 20 = 8a + 8 \). So the expression becomes \( \frac{8a + 8}{a(a+4)(a-4)} \).

Step 5: Simplify the Result

Factor the numerator: \( 8a + 8 = 8(a + 1) \). The expression is now \( \frac{8(a+1)}{a(a+4)(a-4)} \). This is already in the simplest form as there are no common factors to cancel.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying algebraic fractions. It involves expressing a polynomial as a product of its factors, which are simpler polynomials or numbers. For example, in the expression \( \frac{3}{a^2 + 4a} \), the polynomial in the denominator \( a^2 + 4a \) can be factored as \( a(a + 4) \). Here’s how the process typically works:

• Identify Common Factors: Look for numbers or variables that are present in all terms of the polynomial. In \( a^2 + 4a \), the common factor is \( a \).
• Factor by Grouping: This involves grouping terms and finding common factors in each group.
• Check Special Patterns: Recognize patterns like the difference of squares, perfect square trinomials, or the sum/difference of cubes.

Factoring helps in finding a common denominator, making algebraic fractions easier to add, subtract, or simplify. It essentially breaks down complex problems into simpler components that are easier to work with.

Least Common Denominator

The least common denominator (LCD) is pivotal when working with algebraic fractions, especially in operations like addition or subtraction. The LCD is the smallest expression that can serve as a common denominator for two or more fractions. Take for instance the fractions \( \frac{3}{a(a+4)} \) and \( \frac{5}{a(a-4)} \). The denominators \( a(a+4) \) and \( a(a-4) \) have the common factor \( a \). We find the LCD by multiplying all distinct factors:

• Factor each denominator first: \( a(a + 4) \) and \( a(a - 4) \).
• Combine them: The LCD is \( a(a + 4)(a - 4) \).

The LCD allows us to write equivalent fractions with the same denominator, making it possible to perform addition or subtraction operations. Finding the LCD simplifies complex fractions, aligns their denominators, and is essential for combining algebraic expressions accurately.

Simplifying Expressions

Simplifying expressions means reducing an expression to its most basic form without changing its value, resulting in a clearer, more concise expression. For example, the expression \( \frac{8(a+1)}{a(a+4)(a-4)} \) is simplified since no common factors exist to cancel out in both the numerator and the denominator.

• Combine Like Terms: When possible, add or subtract terms that are similar. For example, \( 3(a-4) + 5(a+4) \) simplifies to \( 8a + 8 \).
• Cancel Common Factors: Check for any common factors in the numerator and denominator that can be reduced or canceled. For example, if you had \( \frac{8(a+1)}{8a(b+4)(b-4)} \), the 8's could be canceled.

Simplifying expressions is essential for obtaining the most efficient version of a mathematical expression. It’s important because it often reveals the true nature of the relationship between terms and can provide the simplest solution to a problem.