Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operations and simplify.\(\frac{x}{x^{2}+x-2}-\frac{1}{x+2}\)

Step-by-Step Solution

Answer

The simplified form of \( \frac{x}{x^{2}+x-2} - \frac{1}{x+2} \) is \( \frac{1}{(x-1)(x+2)} \).

Step 1: Factorize both denominators

Starting with this calculation, first factorize the expressions in the denominators where applicable.\(\frac{x}{x^{2}+x-2}-\frac{1}{x+2}\)becomes \(\frac{x}{(x-1)(x+2)}-\frac{1}{x+2}\) after factoring \(x^{2}+x-2\) into \((x-1)(x+2)\).

Step 2: Find common denominator

The next step is to find a common denominator. Since one denominator is a multiple of the other, you can use the larger one \((x-1)(x+2)\) as the common denominator. Thus, the equation becomes \(\frac{x}{(x-1)(x+2)}-\frac{(x-1)}{(x-1)(x+2)}\). The \(1\) in the second fraction was multiplied by \((x-1)\) to become \((x-1)\), which gives us the common denominator.

Step 3: Combine and simplify

Now that the technically challenging parts have been dealt with, all what remains is technicalities. Subtract the two fractions now that they have the same denominators: \(\frac{x-(x-1)}{(x-1)(x+2)} = \frac{1}{(x-1)(x+2)}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill when dealing with rational expressions. It involves rewriting a quadratic expression, generally in the form \(ax^2 + bx + c\), as a product of two binomials.

To factor a quadratic expression like \(x^2 + x - 2\), we need to find two numbers that multiply to the constant term -2 and add to the linear coefficient 1. These numbers are -1 and +2. Thus, the expression can be rewritten as \((x - 1)(x + 2)\).

Here's how you can identify these numbers quickly:

• Look for two numbers that multiply to give the constant term (in this case, -2).
• Ensure the same numbers add up to the middle coefficient (here, +1).

Replacing the quadratic expression with these factors simplifies operations with rational expressions.

Common Denominator

Finding a common denominator is crucial when you need to add or subtract fractions. It helps in aligning the fractions so they can be combined effectively.

In our original exercise, the denominators \((x - 1)(x + 2)\) and \(x + 2\) required a common denominator. Since \((x - 1)(x + 2)\) was a complete factorization of the first denominator, it served as the common denominator for both fractions.

Let's break down the method:

• Identify the larger denominator if one is a factor of the other, use it as the common denominator. Here, \((x - 1)(x + 2)\) encompasses \(x + 2\).
• Rewrite each fraction to have this common denominator. Multiply the numerator of the smaller denominator fraction by the needed factor, in this case, \((x - 1)\).

This ensures both fractions can be directly subtracted or added.

Simplifying Fractions

Simplifying fractions is the final step in rational expressions. Once you have a common denominator, you can combine the fractions.

In the exercise, we subtracted \(\frac{x}{(x - 1)(x + 2)}\) from \(\frac{x-1}{(x - 1)(x + 2)}\). Simplifying this involved aligning numerators and subtracting them:

• Line up numerators: \(x - (x - 1)\).
• Subtract: the negative sign distributes across, transforming into \(x - x + 1\), which simplifies to \(1\).

This results in the simple fraction \(\frac{1}{(x - 1)(x + 2)}\).

Always check if further simplification can occur, such as cancelling out common factors, although in this exercise the result is a simplified rational expression.