Problem 11 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{3}{x^{2}-16}+\frac{5}{x+4} $$

Step-by-Step Solution

Answer

\(\frac{5x - 17}{x^2 - 16}\)

Step 1: Identify the Common Denominator

The fractions have denominators that need to be combined into a common one. The denominators are \(x^2 - 16\) and \(x + 4\). Notice that \(x^2 - 16\) is a difference of squares and can be factored as \((x-4)(x+4)\). This factored form shares a factor with \(x + 4\), so the common denominator will be \((x-4)(x+4)\).

Step 2: Rewrite Each Fraction

Rewrite the fractions with the same common denominator. Multiply the top and bottom of the second fraction by \(x - 4\) to match the common denominator. This gives:\[ \frac{3}{x^2 - 16} = \frac{3}{(x-4)(x+4)} \]\[ \frac{5}{x+4} = \frac{5(x-4)}{(x+4)(x-4)} \]

Step 3: Combine the Fractions

Now that both fractions have the same denominator, you can add them by combining their numerators:\[ \frac{3}{(x-4)(x+4)} + \frac{5(x-4)}{(x-4)(x+4)} = \frac{3 + 5(x-4)}{(x-4)(x+4)} \]

Step 4: Simplify the Numerator

Distribute \(5\) into the expression \(5(x-4)\) to simplify:\[ 5(x-4) = 5x - 20 \]Combine this with the constant term \(3\) in the numerator:\[ \frac{3 + 5x - 20}{(x-4)(x+4)} = \frac{5x - 17}{(x-4)(x+4)} \]

Step 5: Present the Final Answer

Ensure the fraction is in simplest form by verifying any common factors:No further cancelation is possible because \(5x - 17\) shares no common factors with \((x-4)(x+4)\). Therefore, the simplified form is:\[ \frac{5x - 17}{x^2 - 16} \]

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This process can make complex expressions easier to work with and understand.

When simplifying rational expressions, much like numerical fractions, it’s essential first to check for common factors in the numerator and denominator.

• If common factors exist, you can divide both the numerator and the denominator by those factors to reduce the fraction.
• For polynomial fractions, there often are underlying algebraic relationships, such as the difference of squares, which can be factored to help in simplification.

Applying simplification to algebraic fractions is crucial to not only ease computations but also to reveal more insightful properties or limitations of the function represented.

Factoring Polynomials

Factoring polynomials is a method of expressing a polynomial as the product of its factors, which are simpler polynomials. This is an essential skill in algebra because it simplifies expressions and solves equations effectively.

For example, the polynomial expression \(x^2 - 16\) can be factored by recognizing it as a difference of squares:

• The difference of squares states that \(a^2 - b^2 = (a - b)(a + b)\).
• For \(x^2 - 16\), identify \(a^2 = x^2\) and \(b^2 = 16\), thus \((x - 4)(x + 4)\).

Factoring helps to find common denominators in algebraic fractions, solve polynomial equations, and even integrate certain types of functions. Without factoring, many expressions remain cumbersome and challenging to manipulate.

Common Denominator

Finding a common denominator is a critical step when adding or subtracting fractions, particularly when those fractions have distinct denominators. It ensures that the fractions' numerators can be directly added or subtracted.

In any rational expression, to achieve a common denominator:

• Examine each fraction's denominator and determine the least common denominator (LCD) that all denominators can divide into.
• Factor each denominator, if necessary, to clearly identify shared components.
• Multiply the top and bottom of each fraction by necessary factors so each fraction shares the LCD.

For the example \(\frac{3}{x^2 - 16} + \frac{5}{x+4}\), factoring \(x^2 - 16\) simplifies identifying the common denominator of \((x-4)(x+4)\). This technique facilitates the direct addition or subtraction of fractions by aligning their denominators into a single, manageable form.