Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{2 x-3}{x-1}-\frac{2 x+1}{x+1} $$

Short answer

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

Step-by-step solution

- Identify the Common Denominator

The common denominator for the fractions \( \frac{2x-3}{x-1} \) and \( \frac{2x+1}{x+1} \) is \( (x-1)(x+1) \).

- Rewrite Each Fraction with the Common Denominator

Rewrite each fraction with the common denominator. For the first fraction:\[\frac{2x-3}{x-1} = \frac{(2x-3)(x+1)}{(x-1)(x+1)}\]For the second fraction:\[\frac{2x+1}{x+1} = \frac{(2x+1)(x-1)}{(x-1)(x+1)}\]

- Expand Numerators

Expand the numerators of each fraction:\[(2x-3)(x+1) = 2x^2 + 2x - 3x - 3 = 2x^2 - x - 3\]\[(2x+1)(x-1) = 2x^2 - 2x + x - 1 = 2x^2 - x - 1\]

- Subtract the Numerators

Combine the fractions by subtracting the numerators:\[\frac{(2x^2 - x - 3) - (2x^2 - x - 1)}{(x-1)(x+1)}\]Simplify the numerator by eliminating like terms:\[(2x^2 - x - 3) - (2x^2 - x - 1) = -3 + 1 = -2\]

- Write the Simplified Fraction

The simplified fraction is:\[\frac{-2}{(x-1)(x+1)}\]

Key concepts

Common Denominator

Understanding common denominators is vital when working with fractions, especially algebraic fractions. A common denominator is a shared multiple of the denominators of two or more fractions. It allows us to combine the fractions into a single fraction.

In our example, we are working with the fractions \(\frac{2x-3}{x-1}\) and \(\frac{2x+1}{x+1}\). Both denominators, \((x-1)\) and \((x+1)\), need a common multiple. The common denominator is \((x-1)(x+1)\).

We rewrite each fraction to have this common denominator. This step is crucial for correctly performing operations like addition or subtraction with fractions.

Factored Form

Leaving your answer in factored form means presenting it as a product of its factors rather than as an expanded polynomial.

Factored form can simplify the further manipulation and understanding of algebraic expressions. For instance, in our simplified fraction \(\frac{-2}{(x-1)(x+1)}\), the denominator remains in its factored state, which is essential as it clearly shows the roots of the polynomial denominator.

Always check whether the factors are simplified completely, and remain in the right form specified by the problem.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions (expressions involving variables). They follow similar rules as numeric fractions but require additional steps like factoring and finding common denominators.

In our example, both \(\frac{2x-3}{x-1}\) and \(\frac{2x+1}{x+1}\) are algebraic fractions. They include polynomials in the numerators and simple binomial expressions in the denominators. Dealing effectively with algebraic fractions involves understanding variable manipulation, distribution, and combining like terms.

Numerator Expansion

Numerator expansion refers to distributing and combining the terms in the numerator when the fraction has been rewritten to share a common denominator.

In the example, we first rewrite \(\frac{2x-3}{x-1}\) and \(\frac{2x+1}{x+1}\) with the common denominator \((x-1)(x+1)\). From there, we expand the numerators:

• \begin{itemize}
• Expanding \(2x-3)(x+1)\): \2x^2 + 2x - 3x - 3 = 2x^2 - x - 3\
• Expanding \(2x+1)(x-1)\): \2x^2 - 2x + x - 1 = 2x^2 - x - 1\

This step helps combine the fractions by subtracting the expanded numerators and further simplifying the expression. This subtraction simplifies our example to \(\frac{-2}{(x-1)(x+1)}\).