Problem 37 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{2}{x}+\frac{3 x+1}{x^{2}}-\frac{x-2}{x^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5x^2 + 2}{x^3} \).

Step 1: Find a Common Denominator

The expression given is \( \frac{2}{x} + \frac{3x+1}{x^2} - \frac{x-2}{x^3} \). To simplify, we first need a common denominator for all fractions. The denominators are \( x \), \( x^2 \), and \( x^3 \). The least common denominator (LCD) is \( x^3 \).

Step 2: Rewrite Each Fraction with the Common Denominator

Convert each fraction so that they all have the denominator \( x^3 \):- \( \frac{2}{x} \) becomes \( \frac{2 imes x^2}{x^3} = \frac{2x^2}{x^3} \).- \( \frac{3x+1}{x^2} \) becomes \( \frac{(3x+1) imes x}{x^3} = \frac{3x^2 + x}{x^3} \).- \( \frac{x-2}{x^3} \) is already \( \frac{x-2}{x^3} \).

Step 3: Combine the Fractions

Now, combine the fractions into a single expression:\[\frac{2x^2}{x^3} + \frac{3x^2 + x}{x^3} - \frac{x-2}{x^3} = \frac{2x^2 + 3x^2 + x - (x - 2)}{x^3}\]

Step 4: Simplify the Numerator

Simplify the numerator by combining like terms:\[2x^2 + 3x^2 + x - x + 2 = 5x^2 + 2\]

Step 5: Write the Final Expression

Using the simplified numerator, the expression becomes:\[\frac{5x^2 + 2}{x^3}\]

Step 6: Verify the Simplification

Double-check every step to ensure no computational errors were made. Since we started with different denominators and combined the terms correctly, the simplified expression is verified as \( \frac{5x^2 + 2}{x^3} \).

Key Concepts

Common Denominators

When dealing with fractions, especially ones that need to be added or subtracted, finding a common denominator is crucial. The common denominator is essentially a shared base you use to make sure all fractions are comparable and can be combined. Think of it as ensuring everyone speaks the same language in a meeting to communicate effectively.

To find the least common denominator (LCD), you identify the highest power of each variable across all fractions. For the example in the exercise, the denominators are \( x \), \( x^2 \), and \( x^3 \). The LCD in this case is \( x^3 \) because it covers the highest power necessary for any of the fractions involved. This unified format allows for the straightforward combination of the terms. Remember, the process of finding common denominators involves:

• Listing out the denominators.
• Identifying the largest power of each variable present.
• Multiplying these together to form the least common denominator.

Finding the LCD is fundamental to ensure smooth arithmetic operations on fractions.

Simplifying Expressions

Once you have a common denominator, the next step is simplifying the expression. Simplification makes expressions easier to handle and understand. It involves reducing the expression to its most straightforward form, often by unifying terms or canceling out unnecessary components.

In this exercise, after rewriting all fractions with the \( x^3 \) denominator, the expression becomes much simpler to handle. You can then focus on the numerators. Combining like terms is a typical method for simplification, which involves:

• Grouping similar terms together.
• Adding or subtracting the coefficients of these grouped terms.
• Ensuring every term is simplified as much as possible.

This step of simplifying ensures the expression is not only correct but also presented in the clearest and most concise way for anyone reading or working with it.

Fraction Operations

Operations with fractions, especially algebraic ones, involve more than just straightforward addition or subtraction. It requires a keen understanding of how fractions transform and interact with variables.

When adding or subtracting fractions, as seen in the exercise, aligning them under a common denominator is crucial. This not only involves ensuring uniformity but also requires rewriting each fraction carefully while maintaining mathematical accuracy. The next task is then to combine them by operating directly on the numerators without disturbing the denominators:

• Rewritten fractions using the common denominator enable whole expression transformations.
• Operators between fractions (addition or subtraction) are applied directly to the numerators once they share a common base.
• Rearrange the numerators, check for like terms, and perform the necessary arithmetic operations.

These fraction operations solidify the base for more advanced algebraic manipulation, setting the tone for clear and error-free algebra expressions.