Problem 15 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{4}{a^{2}-2 a}+\frac{7}{a^{2}+2 a} $$

Step-by-Step Solution

Answer

\( \frac{11a - 6}{a(a - 2)(a + 2)} \)

Step 1: Factoring the Denominators

The first step is to factor the denominators of the fractions. \( a^2 - 2a \) can be factored as \( a(a - 2) \), and \( a^2 + 2a \) can be factored as \( a(a + 2) \).

Step 2: Finding the Least Common Denominator (LCD)

The Least Common Denominator needs to include all the factors from the factored denominators. In this case, the LCD is \( a(a - 2)(a + 2) \).

Step 3: Rewriting the Fractions with the LCD

Rewrite each fraction with the common denominator:\[ \frac{4}{a(a - 2)} = \frac{4(a + 2)}{a(a - 2)(a + 2)} \]\[ \frac{7}{a(a + 2)} = \frac{7(a - 2)}{a(a - 2)(a + 2)} \].

Step 4: Combine the Fractions

Add the fractions using the common denominator:\[ \frac{4(a + 2) + 7(a - 2)}{a(a - 2)(a + 2)} = \frac{4a + 8 + 7a - 14}{a(a - 2)(a + 2)} \].

Step 5: Simplify the Numerator

Simplify the expression in the numerator:\[ 4a + 8 + 7a - 14 = 11a - 6 \].

Step 6: Write the Final Answer

The simplified expression is:\[ \frac{11a - 6}{a(a - 2)(a + 2)} \].

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which are simpler polynomials. This helps in simplifying expressions, especially when dealing with fractions. When faced with a polynomial in a fraction's denominator, the first task is to factor it.

For example, consider the expressions in the exercise:

• \( a^2 - 2a \) could be rewritten by taking \(a\) as a common factor, resulting in \( a(a - 2) \).
• Similarly, \( a^2 + 2a \) can be factored as \( a(a + 2) \).

Factoring is crucial as it reveals the building blocks of the expression and simplifies further operations. These factors then become essential in determining the least common denominator, leading to a more manageable form of the expression.

Least Common Denominator

Finding the least common denominator (LCD) is a key step when working with algebraic fractions. The LCD lets you combine fractions by providing a common base for comparison and ease of operation. To determine the LCD, identify all unique factors from each denominator.

For the exercise at hand:

• The denominators are \( a(a - 2) \) and \( a(a + 2) \).
• The LCD must include each unique factor: \( a \), \( a - 2 \), and \( a + 2 \).
• Therefore, the LCD here is \( a(a - 2)(a + 2) \).

With the LCD, you can rewrite each fraction to have this common denominator, facilitating the addition or subtraction of the algebraic fractions.

Simplifying Fractions

Simplifying fractions aims to reduce a fraction into its simplest form by eliminating any common factors between the numerator and the denominator. After obtaining the common denominator, the next steps involve operations on the numerators.

Let's break down the steps as seen in the exercise:

• Reformulate each fraction with the least common denominator, which results in adjusting the numerators. For example, \( \frac{4}{a(a-2)} \) becomes \( \frac{4(a+2)}{a(a-2)(a+2)} \).
• Add or subtract the numerators as indicated: \( 4(a+2) + 7(a-2) \) simplifies to \( 11a - 6 \).
• Finally, the expression \( \frac{11a - 6}{a(a-2)(a+2)} \) is the simplified form as the numerator and the different factors in the denominator do not share common factors.

Simplifying fractions helps in clearer understanding and further manipulation of the expression, making it tidier and often more useful for further calculations.