Problem 69 - Algebra and Trigonometry

Question

simplify each complex rational expression. $$ \frac{\frac{1}{x+1}}{\frac{1}{x^{2}-2 x-3}+\frac{1}{x-3}} $$

Step-by-Step Solution

Answer

\(\frac{x(x-3)}{x^2 - 3x + 4}.\)

Step 1: Simplify the Denominator

To simplify the denominator, first find a common denominator for the two fractions, which is \(x^2-2x-3\) and \(x-3\). The common denominator would be \(x(x-3)\). Now rewrite the fractions in the denominator with the common denominator: \(\frac{1}{x^2 - 2x - 3} + \frac{1}{x-3} = \frac{1}{(x-1)(x+3)} + \frac{(x+1)}{x(x-3)}\).

Step 2: Combine the Fractions in the Denominator

Now add these two fractions and simplify: \(\frac{1}{(x-1)(x+3)} + \frac{(x+1)}{x(x-3)} = \frac{x+1+(x-1)(x+3)}{x(x-3)} = \frac{x + 1 + x^2 - 4x + 3}{x(x - 3)} = \frac{x^2 - 3x + 4}{x(x - 3)}\)

Step 3: Simplify the Whole Expression

Lastly, simplify the whole expression by replacing the denominator with the expression found in step 2: \(\frac{\frac{1}{x+1}}{\frac{x^2 - 3x + 4}{x(x - 3)}} = \frac{x(x-3)}{x^2 - 3x + 4}\). This is the simplified form of the complex rational expression.

Key Concepts

Understanding the Common Denominator

When simplifying complex rational expressions, finding a common denominator is crucial. This concept is similar to finding a common language for communication; it's the basis that allows different parts to come together seamlessly. In algebra, to combine fractions effectively, we must first rewrite them so they share the same denominator. This process involves identifying the least common multiple (LCM) of the denominators. In our case, with denominators \(x^2-2x-3\) and \(x-3\), we can observe that these expressions can factorize and the LCM of \(x - 3\) and \(x - 1\) is indeed \(x(x - 3)\).

Once identified, we rewrite each fraction by multiplying its numerator and denominator by whatever is needed to achieve the common denominator. This allows us to add or subtract the fractions later. As a tip, always check the original denominators for factors that might simplify the process of finding a common denominator, such as shared variables or numbers.

Combining Fractions with Algebra

The next step, combining fractions, is where the common denominator comes into play. Think of this process as bringing together slices of a pie to make a whole; we're trying to put together the parts of an algebraic expression. To do this, we keep the common denominator and sum up the numerators. The resulting fraction is a single entity that represents the sum (or difference) of the original fractions.

In our exercise, after finding a common denominator, we combine \(\frac{1}{(x-1)(x+3)}\) and \(\frac{(x+1)}{x(x-3)}\) by adding their numerators, while the common denominator remains unchanged. An essential part of this step is to simplify the numerator by combining like terms and factoring if possible. This might lead to further simplification with the common denominator and result in a more straightforward expression. Combining fractions sounds straightforward, but keep in mind that attention to detail is vital when working with algebraic expressions to avoid errors.

Simplifying Algebraic Expressions

The finishing touch in dealing with complex rational expressions is simplifying algebraic expressions. Simplification can involve reducing fractions, canceling out common factors in the numerator and denominator, or factoring and simplifying polynomial expressions. Think of simplification as the art of tidying up: we want the expression to be as neat and efficient as possible without altering its value.

To achieve this, inspect the numerators and denominators for common factors, and reduce wherever you can. When faced with a complex fraction, like in our example, you can simplify by multiplying the numerator of the main fraction by the reciprocal of its denominator. The end goal is always to reach the simplest form possible, ideally where no further factoring or reduction can occur. Remember that simplifying an expression doesn’t just make it easier to read; it can also reveal insights into the properties of the expression and how it behaves within a given context.