Problem 45 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\frac{3x + 2}{(x+1)^2}\)

Step 1: Identify the Common Denominator

Given the two fractions \(\frac{x}{(x+1)^2}\) and \(\frac{2}{x+1}\), we need a common denominator to combine them. The first denominator is \((x+1)^2\), and the second is \(x+1\). The least common denominator is \((x+1)^2\), since it is already the square form of the second denominator.

Step 2: Rewrite the Second Fraction

We need to rewrite the second fraction \(\frac{2}{x+1}\) so it has the common denominator \((x+1)^2\). Multiply both the numerator and denominator of \(\frac{2}{x+1}\) by \(x+1\) to get \(\frac{2(x+1)}{(x+1)^2}\).

Step 3: Perform the Addition

Now that both fractions \(\frac{x}{(x+1)^2}\) and \(\frac{2(x+1)}{(x+1)^2}\) have a common denominator, combine them into a single fraction: \(\frac{x + 2(x+1)}{(x+1)^2}\).

Step 4: Simplify the Numerator

Expand and simplify the numerator: \(x + 2(x + 1) = x + 2x + 2 = 3x + 2\). Now the fraction is \(\frac{3x + 2}{(x+1)^2}\).

Step 5: Review for Further Simplification

Check for possible factorization or simplifications in the numerator \(3x + 2\) and the denominator \((x+1)^2\). In this case, there are no common factors, so the expression is already in its simplest form.

Key Concepts

Common Denominator

When you're dealing with the addition or subtraction of fractions, having a common denominator is crucial. The denominator is the number below the fraction's line, and fractions must share the same denominator to be easily added or subtracted. In our example, we have the fractions \(\frac{x}{(x+1)^2}\) and \(\frac{2}{x+1}\). To find a common denominator, observe both denominators closely:

• First denominator: \((x+1)^2\)
• Second denominator: \(x+1\)

The least common denominator in this case is \((x+1)^2\) because it incorporates the whole of the second denominator, \(x+1\), raised to the power of two. To combine fractions, both must effectively convert (or already have) their existing denominators into this common denominator. By adjusting the numerators accordingly, the fractions can then be added together seamlessly.

Simplification

Once you have a common denominator, focus shifts to creating a neat and simplified result. After combining fractions under a common denominator, you get a new expression that often needs cleaning up. In our problem, the expression becomes \(\frac{x + 2(x+1)}{(x+1)^2}\) after finding a common denominator.
To simplify, begin by expanding the numerator:

• Expand: \(x + 2(x + 1)\)
• Calculate: This becomes \(x + 2x + 2\) which simplifies further to \(3x + 2\)

Thus, the fraction is simplifed to \(\frac{3x + 2}{(x+1)^2}\).
Simplify as much as possible by checking for common factors in the numerator and denominator. In this case, since \(3x + 2\) and \( (x+1)^2 \) share no common factors, the expression is already in its simplest form.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. In simple terms, they are just like fractions you are familiar with, but with expressions, variable terms, and possibly higher powers. In this example, we handled the rational expressions \(\frac{x}{(x+1)^2}\) and \(\frac{2}{x+1}\).
The operations we performed—finding a common denominator, adjusting numerators, and simplifying—are typical tasks when working with rational expressions.

• First, identify the structure, focusing on the polynomials involved.
• Then, operate using familiar fraction rules: common denominators and simplification.

Rational expressions can sometimes be complex due to the presence of variables and their interactions, but by tackling them step-by-step, as shown in our example, they can be manageable. Treat them as you would regular fractions, remembering that practice and patience are key to mastering them.