Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ f(x)=\frac{x^{6}-x}{x^{3}} $$

Step-by-Step Solution

Answer

The simplified function is \( f(x) = x^3 - x^{-2} \).

Step 1: Simplify the Expression

The function given is \( f(x) = \frac{x^6 - x}{x^3} \). To simplify this expression, we can divide both terms in the numerator by the denominator \( x^3 \). This gives: \[ f(x) = \frac{x^6}{x^3} - \frac{x}{x^3} \].

Step 2: Simplify Each Term Separately

Simplify each term separately:- \( \frac{x^6}{x^3} = x^{6-3} = x^3 \)- \( \frac{x}{x^3} = x^{1-3} = x^{-2} \)Thus, the simplified expression is: \[ f(x) = x^3 - x^{-2} \].

Step 3: Write the Final Simplified Expression

Combine the simplified terms together to express the function in its most reduced form: \( f(x) = x^3 - x^{-2} \).

Key Concepts

Polynomial Division

Polynomial division is a key technique used in simplifying rational expressions, such as the one you've encountered in the exercise. When you're faced with a fraction that involves polynomials both in the numerator and the denominator, you should aim to divide each term in the numerator separately by the term in the denominator. The goal is to simplify the entire expression by breaking it down into more manageable parts.

In the original problem, you have the polynomial in the numerator, which is expressed as \(x^6 - x\), and you're dividing it by \(x^3\). You divide each term of the polynomial individually.

• First, divide \(x^6\) by \(x^3\)
• Then, divide \(x\) by \(x^3\)

Once broken down, this technique allows you to apply exponent rules effectively in the next steps of the solution. Remember, polynomial division helps to clearly see how each term simplifies, providing a neat pathway to the final result.

Exponent Rules

Exponent rules are fundamental when dealing with polynomial division, especially in simplified expressions. In this exercise, they are applied to reduce the expression into its simplest form.

The primary rule we are applying here is: when dividing like bases (same base number), subtract their exponents. For example:

• If you have \(\frac{x^6}{x^3}\), you apply the rule \(x^{a-b} = x^{6-3} = x^3\)
• For \(\frac{x^1}{x^3}\), it becomes \(x^{1-3} = x^{-2}\)

These rules allow you to take a more complex expression and break it down into simpler parts with positive or negative exponents.

Understanding these rules provides clarity and speed when simplifying expressions in any mathematical problem solving scenario—making algebraic tasks simpler and more intuitive.

Mathematical Problem Solving

Mathematical problem solving involves several steps and strategies to work through problems effectively. It's about understanding the problem, deciding on a strategy, and then carrying it out. In the context of the given exercise, the problem-solving process involved simplifying a rational expression step-by-step.

First, identify what the problem is asking. Here, it's asking to simplify \( f(x) = \frac{x^6 - x}{x^3} \). The strategy used is polynomial division combined with exponent rules.

Solving such problems enhances your skills in approaching algebraic expressions sensibly. You proceed by simplifying term by term, making complex polynomials easier to manage. Problem-solving isn’t just about finding the correct answer but also involves understanding why a solution works.

• Identify the type of operations needed (e.g., division, subtraction of exponents)
• Use known rules and formulas effectively (e.g., polynomial division, exponent rules)
• Verify the simplified expression

Lastly, always double-check your work. This way, you ensure the accuracy and completeness of your solution, enhancing your mathematical intuition and confidence.