Problem 102 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{3}{x-5}-\frac{1}{x-3}-\frac{2 x}{x-5} $$

Step-by-Step Solution

Answer

\(\frac{-2x^2 + 8x - 14}{(x-5)(x-3)}\)

Step 1: Identify the Problem Type

We are given a rational expression with three fractions to simplify. The first step is identifying that we need a common denominator to combine these terms.

Step 2: Determine the Common Denominator

The denominators are \(x-5\), \(x-3\), and \(x-5\). The least common denominator for these expressions is \((x-5)(x-3)\).

Step 3: Rewrite Each Fraction

Rewrite each fraction with the common denominator \((x-5)(x-3)\):- \(\frac{3}{x-5}\) becomes \(\frac{3(x-3)}{(x-5)(x-3)}\)- \(\frac{1}{x-3}\) becomes \(\frac{1(x-5)}{(x-5)(x-3)}\)- \(\frac{2x}{x-5}\) becomes \(\frac{2x(x-3)}{(x-5)(x-3)}\)

Step 4: Combine the Fractions

Combine the rewritten fractions into a single fraction:\[\frac{3(x-3) - 1(x-5) - 2x(x-3)}{(x-5)(x-3)}\]

Step 5: Simplify the Numerator

Simplify the expression in the numerator:- Distribute each term: - \(3(x-3) = 3x - 9\) - \(1(x-5) = x - 5\) - \(2x(x-3) = 2x^2 - 6x\)- Combine the terms: - \(3x - x + 5 - 2x^2 + 6x = -2x^2 + 8x - 14\)

Step 6: Rewrite the Simplified Expression

The simplified expression becomes:\[\frac{-2x^2 + 8x - 14}{(x-5)(x-3)}\]

Key Concepts

Common Denominator

When working with rational expressions, finding a common denominator is crucial, much like when dealing with regular fractions. A common denominator means choosing a common multiple for the denominators of the fractions involved. This allows us to combine or simplify the fractions. To find a common denominator, we often take the least common multiple (LCM) of the denominators.
In this case, for the expressions \(\frac{3}{x-5}\) and \(\frac{1}{x-3}\), the denominators \(x-5\) and \(x-3\) dictate that the LCM is \((x-5)(x-3)\). This is the simplest form that can accommodate both denominators.

• Identify the unique factors in each denominator.
• Multiply these factors together to get the common denominator.

Once you have a common denominator, rewrite each fraction so that they all share this common base. This step enables seamless addition or subtraction across fractions, which is essential in simplifying them.

Simplify Fractions

Simplifying fractions involves rewriting them in their simplest forms. For rational expressions, this often requires rewriting complex fractions with a common denominator, and then simplifying their numerators.
Once fractions have a common denominator, their numerators can be combined. For instance, with the fractions \(\frac{3(x-3)}{(x-5)(x-3)}\), \(\frac{1(x-5)}{(x-5)(x-3)}\), and \(\frac{2x(x-3)}{(x-5)(x-3)}\), we focus on simplifying their numerators:

• Apply the distributive property to expand expressions.
• Combine like terms to streamline the overall expression.

By distributing and combining like terms correctly, the expression \(\frac{-2x^2 + 8x - 14}{(x-5)(x-3)}\) emerges from complex simplification processes. This demonstrates the importance of correct distribution and combination in simplifying fractions.

Polynomial Expressions

Polynomial expressions are key components of rational expressions, as they often appear in numerators and denominators. These expressions contain variables raised to powers combined with coefficients, expressed generally in forms like \(ax^2 + bx + c\).
When simplifying rational expressions, polynomials in numerators or as part of finding a common denominator commonly appear. In the step-by-step solution, for example, each rational component involves manipulations of polynomials:

• The fraction \(3(x-3)\) becomes polynomial \(3x - 9\).
• The term \(2x(x-3)\) expands to \(2x^2 - 6x\).

Understanding polynomial operations, including addition, subtraction, and factoring, is critical to simplifying these expressions efficiently.