Problem 56 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{27}{y^{2}-81}+\frac{3}{2(y+9)}\)

Step-by-Step Solution

Answer

\(\frac{3}{2(y-9)}\)

Step 1: Factor the Denominator

First, we need to factor each denominator that is factorizable. The denominator of the first fraction is \(y^2 - 81\), which is a difference of squares. So, we can factor it as \((y-9)(y+9)\). The denominator of the second fraction is already factored as \(2(y+9)\).

Step 2: Determine the Least Common Denominator (LCD)

To add these fractions, we need a common denominator. The denominators after factoring are \((y-9)(y+9)\) and \(2(y+9)\). The least common denominator (LCD) is \(2(y-9)(y+9)\).

Step 3: Rewrite Each Fraction with the LCD

Rewrite each fraction so that both have the LCD. The first fraction becomes \(\frac{27}{(y-9)(y+9)}\cdot \frac{2}{2}\) which simplifies to \(\frac{54}{2(y-9)(y+9)}\). The second fraction \(\frac{3}{2(y+9)}\) becomes \(\frac{3(y-9)}{2(y-9)(y+9)}\) after multiplying the numerator and denominator by \((y-9)\).

Step 4: Combine the Fractions

Now that both fractions have the same denominator, we can combine them: \(\frac{54}{2(y-9)(y+9)} + \frac{3(y-9)}{2(y-9)(y+9)} = \frac{54 + 3(y-9)}{2(y-9)(y+9)}\).

Step 5: Simplify the Numerator

Distribute and simplify the numerator: \(54 + 3(y - 9) = 54 + 3y - 27 = 3y + 27\).

Step 6: Combine and Simplify Further, If Possible

Combine the simplified terms: \(\frac{3y + 27}{2(y-9)(y+9)}\). Factor the numerator \(3y + 27\) to \(3(y + 9)\). The fraction becomes \(\frac{3(y + 9)}{2(y-9)(y+9)}\). The terms \((y+9)\) cancel out, giving \(\frac{3}{2(y-9)}\).

Key Concepts

Factoring Polynomials

When dealing with rational expressions, factoring polynomials is often an important first step. This is because factoring helps in simplifying the expressions and finding common denominators more easily. Let's consider the first expression: \( \frac{27}{y^2 - 81} \). The denominator \( y^2 - 81 \) is a special type of polynomial known as a difference of squares. A difference of squares can be factored using the formula \( a^2 - b^2 = (a-b)(a+b) \). In this example, 81 can be written as \( 9^2 \), so this becomes \( (y-9)(y+9) \).

• Look for patterns like difference of squares or common factors.
• Breaking down complex expressions makes them easier to work with.
• Factoring is often a crucial step towards simplifying rational expressions.

For the second fraction in the problem: \( \frac{3}{2(y+9)} \), the denominator \( 2(y+9) \) is already in its simplest factored form, so no further action is required. Remember, a factored form will allow you to identify common denominators more easily, simplifying the addition process.

Least Common Denominator

Adding rational expressions, much like adding fractions, requires that both terms share a common denominator. The least common denominator (LCD) is the smallest expression that can be found which both denominators can divide evenly into. In the problem \( \frac{27}{(y-9)(y+9)} + \frac{3}{2(y+9)} \), after factoring, we have denominators of \((y-9)(y+9)\) and \(2(y+9)\).

• The LCD needs to include every factor that appears in any of the individual denominators.
• Consider each distinct factor and take the highest power that appears in any of the denominators.

Thus, the LCD here is \(2(y-9)(y+9)\). It includes both the \((y-9)\) from the first fraction and the \(2\) as well as \((y+9)\) factors that they both share.
Finding the correct LCD is crucial as it ensures that the fractions can be properly combined.

Simplifying Expressions

Simplifying expressions is the final step to make the result as concise as possible. Once the two fractions have a common denominator, they can be added together. In the expression \( \frac{54}{2(y-9)(y+9)} + \frac{3(y-9)}{2(y-9)(y+9)} \), the like terms can be combined into one fraction:

• Combine the numerators while keeping the common denominator fixed.
• Always simplify the resulting expression for neatness and accuracy.

The combined numerator becomes \(54 + 3(y - 9)\). Distribute and combine like terms: \(54 + 3y - 27 = 3y + 27\). The numerator \(3y + 27\) can further be factored to \(3(y + 9)\). Then the expression becomes \(\frac{3(y + 9)}{2(y-9)(y+9)}\). By canceling out the \((y+9)\) terms from the numerator and the denominator, the simplified expression is \(\frac{3}{2(y-9)}\). This final form is crucial because it gives the simplest representation of the expression, making it easier to understand and work with in future calculations.