Problem 58 - Precalculus Mathematics for Calculus

Question

Perform the addition or subtraction and simplify. $$\frac{1}{x+1}-\frac{2}{(x+1)^{2}}+\frac{3}{x^{2}-1}$$

Step-by-Step Solution

Answer

\(\frac{2(x+2)}{(x+1)^2(x-1)}\)

Step 1: Analyze the Denominators

The denominators are \(x+1\), \((x+1)^2\), and \(x^2-1\). Notice that \(x^2-1\) can be factored as \((x+1)(x-1)\). The common denominator for these fractions is \((x+1)^2(x-1)\).

Step 2: Rewrite the Fractions with a Common Denominator

Convert each fraction so that they share the common denominator \((x+1)^2(x-1)\).- \(\frac{1}{x+1}\) must be multiplied by \((x+1)(x-1)\) in the numerator and denominator, giving \(\frac{(x-1)}{(x+1)^2(x-1)}\).- \(\frac{2}{(x+1)^2}\) must be multiplied by \((x-1)\) in the numerator and denominator, becoming \(\frac{2(x-1)}{(x+1)^2(x-1)}\).- \(\frac{3}{x^2-1}\) must be multiplied by \((x+1)\) in the numerator and denominator, resulting in \(\frac{3(x+1)}{(x+1)^2(x-1)}\).

Step 3: Combine the Fractions

Now that all the fractions have the same denominator, combine them:\[\frac{x-1}{(x+1)^2(x-1)} - \frac{2(x-1)}{(x+1)^2(x-1)} + \frac{3(x+1)}{(x+1)^2(x-1)}\].Combine the numerators:\[x - 1 - 2(x - 1) + 3(x + 1)\].

Step 4: Calculate the Combined Numerator

Simplify the expression in the numerator:- Combine like terms: \(x - 1 - 2x + 2 + 3x + 3\).- Simplify: \((x - 2x + 3x) + (-1 + 2 + 3)\).This becomes:\[2x + 4\].

Step 5: Simplify the Expression

Insert the simplified numerator into the fraction:\[\frac{2x + 4}{(x+1)^2(x-1)}\].Check for any further simplification:Factor out a \(2\) from the numerator:\[\frac{2(x+2)}{(x+1)^2(x-1)}\].This is the simplified final expression.

Key Concepts

Common Denominator

When adding or subtracting fractions, finding a common denominator is crucial. It ensures that the fractions are expressed in terms of the same base. This similarity allows for straightforward addition or subtraction of the numerators. In our exercise, the fractions initially had different denominators: \(x+1\), \((x+1)^2\), and \(x^2-1\). We needed to identify a common denominator that could encompass all of these. By factoring \(x^2-1\) into \((x+1)(x-1)\), we determined that the least common denominator was \((x+1)^2(x-1)\).

• This approach entails considering the highest power of each factor that appears in any denominator.
• Once found, each fraction is rewritten so it shares this common denominator.
• This process involves multiplying each fraction by necessary terms to obtain equivalent fractions with a common base.

This solid foundation is indispensable for the correct and simple addition or subtraction of fractions, especially when dealing with variables.

Rational Expressions

Rational expressions resemble fractions but involve polynomials in both the numerator and the denominator. They often arise in algebra when dealing with expressions involving variable terms over other variable terms. In the exercise, expressions like \(\frac{1}{x+1}\) and \(\frac{3}{x^2-1}\) are classic examples of rational expressions.

Combining rational expressions involves several steps:

• Identifying a common denominator, as discussed earlier.
• Adjusting the numerators to reflect this new denominator.
• Adding or subtracting the numerators for combined or simplified expressions.

A key understanding here is that rational expressions can behave like numbers, but with the added complexity of variable terms. Thus, they require careful manipulation, guided by algebraic rules, to ensure that nonsensical results—like division by zero—are avoided. By keeping expressions equivalent, we maintain the mathematical integrity necessary for solving algebraic problems.

Algebraic Simplification

Simplifying algebraic expressions is a fundamental skill in algebra, reducing expressions to their simplest form while preserving their values. During simplification, terms are combined and factored, among other operations, to streamline the expression. In the problem's solution, once the common denominator allowed us to combine the fractions, the next task was to simplify the resulting expression.

Here's how the simplification process unfolded:

• Combine like terms in the numerator: \(x - 1 - 2(x - 1) + 3(x + 1)\).
• Simplify the arithmetic: group variables and constant numbers separately.
• The simplified numerator becomes \(2x + 4\).
• Further streamline by factoring out common factors, resulting in \(\frac{2(x+2)}{(x+1)^2(x-1)}\).

Simplification makes expressions more manageable and often highlights features of the algebraic expression that aren't immediately evident. It may also reveal opportunities for further operations, such as solving for roots or transformation into standard forms.