Problem 35 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{2}{3 s+1}-\frac{9}{(3 s+1)^{2}}$$

Step-by-Step Solution

Answer

\( \frac{6s - 7}{(3s+1)^2} \)

Step 1: Identify the Common Denominator

Both fractions have denominators that involve \( 3s + 1 \). The first fraction has \( 3s + 1 \) and the second has \( (3s+1)^2 \). The common denominator is \( (3s+1)^2 \).

Step 2: Rewrite Fractions with the Common Denominator

Multiply the numerator and denominator of the first fraction \( \frac{2}{3s+1} \) by \( 3s + 1 \) to use the common denominator:\[ \frac{2}{3s+1} = \frac{2(3s+1)}{(3s+1)(3s+1)} = \frac{2(3s+1)}{(3s+1)^2} \]

Step 3: Subtract the Fractions

Now subtract the two fractions:\[ \frac{2(3s+1)}{(3s+1)^2} - \frac{9}{(3s+1)^2} = \frac{2(3s+1) - 9}{(3s+1)^2} \]

Step 4: Simplify the Numerator

Distribute the \(2\) in the expression and simplify:\[ 2(3s + 1) - 9 = 6s + 2 - 9 = 6s - 7 \]

Step 5: Write the Final Simplified Expression

The simplified form of the given expression is:\[ \frac{6s - 7}{(3s+1)^2} \]

Key Concepts

Common Denominator

To simplify any algebraic expression involving fractions, finding a common denominator is often the first step. A common denominator is a shared multiple of the denominators of the fractions involved.
This step ensures all the fractions in the expression can be merged seamlessly.
Understanding how to find a common denominator is crucial in handling algebraic fractions:

• Observe the denominators: Here, we have \(3s + 1\) and \((3s+1)^2\).
• Determine the least common multiple (LCM): In this case, it's \((3s+1)^2\), because the second denominator is the square of the first.

With the common denominator identified, each fraction can be rewritten to share this denominator. This step prepares us to perform operations like addition or subtraction.
Essentially, common denominators remove the obstacle of disconnected fractions and pave the way for simplification.

Fraction Subtraction

Subtracting fractions involves rewriting them with a common denominator, then combining them into one fraction.
Once you have a common denominator, subtracting fractions becomes a straightforward process.
Here's how it works:

• Align denominators: Ensure all fractions have the same denominator. Here, it was achieved using \((3s+1)^2\).
• Subtract numerators: Keep the common denominator and subtract the numerators. For our example, from \(\frac{2(3s+1)}{(3s+1)^2}\) subtract \(\frac{9}{(3s+1)^2}\) leading to \(\frac{2(3s+1) - 9}{(3s+1)^2}\).

The subtraction of numerators, while maintaining the common denominator, allows the fractions to merge into a single algebraic expression. This step is what simplifies the process, turning two fractions into one contiguous expression ready for further simplification.

Polynomial Simplification

Simplifying polynomials is the heart of reducing algebraic expressions to their simplest form. Once you have a single fraction, the numerator often requires simplification.In our example, the steps to simplify the numerator are:

• Distribute: Multiply the term outside the parentheses by each term inside, \(2(3s + 1) = 6s + 2\).
• Subtract constant terms: From \(6s + 2\), you subtract 9, resulting in \(6s - 7\).

This process of distribution and simplification ensures that the expression becomes as concise as possible without losing its original value or characteristics.
Once simplified, the expression stands clearly as a single fraction: \(\frac{6s - 7}{(3s+1)^2}\). This approach not only streamlines polynomials but also unveils underlying patterns or factors that may not be immediately visible. Hence, understanding polynomial simplification is a key aspect of mastering algebra.