Problem 50 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Terms

First, identify the terms to be combined through addition. In this exercise, we have three fractions: \(\frac{1}{x}\), \(\frac{1}{x^{2}}\), and \(\frac{1}{x^{3}}\).

Step 2: Find a Common Denominator

To add the fractions, we need a common denominator. The denominators are \(x\), \(x^2\), and \(x^3\). The least common denominator (LCD) will be \(x^3\), as it is the smallest power that all denominators can divide into.

Step 3: Rewrite Fractions with Common Denominator

Convert each fraction so that they all have the common denominator of \(x^3\):- \(\frac{1}{x} = \frac{x^2}{x^3}\)- \(\frac{1}{x^2} = \frac{x}{x^3}\)- \(\frac{1}{x^3} = \frac{1}{x^3}\)

Step 4: Add the Fractions

Now that all fractions have the same denominator, add them together:\[\frac{x^2}{x^3} + \frac{x}{x^3} + \frac{1}{x^3} = \frac{x^2 + x + 1}{x^3}\]

Step 5: Simplify the Result

The fraction \(\frac{x^2 + x + 1}{x^3}\) is already in its simplest form, as there are no common factors in the numerator and the denominator.

Key Concepts

Common Denominator

When dealing with algebraic fractions, especially during addition or subtraction, finding a common denominator is crucial. The common denominator allows you to combine fractions into a single expression.
In this exercise, we have fractions with different denominators: \(x\), \(x^2\), and \(x^3\). The best way to handle this is to find the Least Common Denominator (LCD), which is the smallest expression that each original denominator can divide into evenly.
The denominators here are powers of \(x\), and among them, \(x^3\) is the greatest power and it can be divided evenly by all three denominators. Thus, \(x^3\) serves as the common denominator for our operation. By transforming each fraction to have this common denominator, we ensure that the addition process is straightforward and accurate.

Fraction Addition

Once all the fractions share a common denominator, the process of adding them becomes much simpler. Think of fractions like parts of a whole.
If all parts are the same size (meaning they have the same denominator), you can easily add the numerators together, just like adding whole numbers.

• The first fraction \(\frac{1}{x}\) becomes \(\frac{x^2}{x^3}\) once multiplied by \(\frac{x^2}{x^2}\).
• The second fraction \(\frac{1}{x^2}\) turns into \(\frac{x}{x^3}\) by multiplying by \(\frac{x}{x}\).
• The third fraction \(\frac{1}{x^3}\) is already expressed with the common denominator.

Now that all fractions have \(x^3\) as the common denominator, simply add the numerators: \(x^2 + x + 1\). The resulting expression \(\frac{x^2 + x + 1}{x^3}\) combines all parts into a singular fraction.

Simplification

Simplification is the final step after performing any operations on algebraic fractions. It involves reducing the fraction to its simplest form. In this instance, the operation has been completed with the fraction \(\frac{x^2 + x + 1}{x^3}\).
To determine if further simplification is possible, inspect both the numerator and the denominator:

• The numerator \(x^2 + x + 1\) has terms that do not share any common factor that also divides the denominator \(x^3\) evenly.

Because there are no common factors, \(\frac{x^2 + x + 1}{x^3}\) is considered fully simplified. Remember, simplification often involves canceling out redundant factors, but here, no such reductions apply. Keeping expressions in their simplest form makes them easier to understand and work with, particularly in more complex algebraic operations.