Problem 17 - 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 \( f(x) = x^3 - \frac{1}{x^2} \).

Step 1: Identify the Function

The given function is \( f(x) = \frac{x^{6}-x}{x^{3}} \). This is a rational function with a numerator \( x^6 - x \) and a denominator \( x^3 \).

Step 2: Factor the Numerator

We need to simplify the expression by factoring. The numerator \( x^6 - x \) can be factored by first factoring out \( x \), resulting in \( x(x^5 - 1) \).

Step 3: Simplify the Expression

Divide both the factored numerator and the denominator by their common factors. The function becomes \( f(x) = \frac{x(x^5 - 1)}{x^3} \), which can be simplified by cancelling \( x \) from the numerator and a part of the denominator. Thus, we get \( f(x) = \frac{x^5 - 1}{x^2} \).

Step 4: Write the Final Simplified Expression

The final simplified form of the function is \( f(x) = x^3 - \frac{1}{x^2} \) after dividing each term of the expression \( \frac{x^5 - 1}{x^2} \) by \( x^2 \).

Key Concepts

Function Simplification

Function simplification is essentially the process of rewriting a function in a simpler or more concise manner without changing its value. It's akin to simplifying fractions in arithmetic. For rational functions, this usually involves either canceling common factors or performing algebraic manipulation to achieve a reduced form.

In the context of our original exercise, simplification plays a critical role. The function given was \( f(x) = \frac{x^6 - x}{x^3} \). Through simplification, the function is made easier to understand and work with, transforming complex expressions into more manageable ones.

• Identify common factors in numerator and denominator.
• Cancel these common factors to simplify the expression.
• Ensure the expression is in its simplest form.

In this case, the simplification process revealed a reduced form: \( f(x) = x^3 - \frac{1}{x^2} \). This shows the power of simplifying as it not only reduces the complexity but also makes further calculations much easier.

Factoring in Calculus

Factoring is a mathematical technique used to break down expressions into multiples or products of simpler expressions. This is a crucial component in calculus, especially when dealing with complex rational functions or polynomial expressions.

Factoring in calculus allows you to simplify expressions and solve equations that would otherwise be too cumbersome to handle directly. For the expression in our exercise, \( x^6 - x \), the first step was factoring. Here, we extracted \( x \) as a common factor resulting in: \( x(x^5 - 1) \).

• Look for the greatest common factor (GCF) to factor out.
• Apply factoring techniques like factoring by grouping, difference of squares, or special polynomial forms as needed.
• Simplify each part to further reduce the expression.

In calculus, factoring is often the key to unlocking more complex problems, as it reveals hidden structures that aren't apparent in their original form. This makes it easier to apply various calculus methods for further analysis.

Polynomial Division

Polynomial division is a technique similar to long division but applied to polynomials instead of numbers. It is an essential tool when working with rational functions, enabling us to simplify them and find aspects like zeros or asymptotes.

In the given problem, polynomial division was applied in the form of simplifying \( \frac{x^5 - 1}{x^2} \). By dividing each term of the polynomial in the numerator separately by the denominator, it results in the simplified expression: \( x^3 - \frac{1}{x^2} \). Breaking down polynomial expressions into simpler terms allows us to more easily interpret and work with the function.

• Align your polynomials by like terms before performing division.
• Use long division or synthetic division as needed.
• Continue division until the remainder is less than the degree of the divisor.

Polynomial division is especially useful for finding horizontal and oblique asymptotes in calculus. With this understanding, students can better analyze the behavior of polynomial functions both graphically and analytically.