Problem 59 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{\frac{1}{(x+h)^{3}}-\frac{1}{x^{3}}}{h}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Expression

The expression given is a complex fraction which needs to be simplified. It involves a difference of fractions in the numerator divided by a single term in the denominator.

Step 2: Find a Common Denominator

To simplify the numerator \( \frac{1}{(x+h)^3} - \frac{1}{x^3} \), we need a common denominator. The common denominator is \( (x+h)^3 x^3 \).

Step 3: Rewrite as a Single Fraction

Rewrite each fraction with the common denominator: \[ \frac{x^3}{(x+h)^3 x^3} - \frac{(x+h)^3}{x^3 (x+h)^3} = \frac{x^3 - (x+h)^3}{x^3 (x+h)^3} \]

Step 4: Substitute Back into the Original Expression

Now substitute the simplified numerator back into the original expression, resulting in: \[ \frac{x^3 - (x+h)^3}{h (x^3 (x+h)^3)} \]

Step 5: Expand \((x+h)^3\) and Simplify

Expand \((x+h)^3\) to \(x^3 + 3x^2h + 3xh^2 + h^3\), then simplify the numerator:\[ x^3 - (x^3 + 3x^2h + 3xh^2 + h^3) = - 3x^2h - 3xh^2 - h^3 \]

Step 6: Factor Out \(h\) from the Numerator

Factor \(h\) from the expanded numerator:\[ -h(3x^2 + 3xh + h^2) \]

Step 7: Cancel \(h\) in the Expression

The expression becomes:\[ \frac{-h(3x^2 + 3xh + h^2)}{h (x^3 (x+h)^3)} \]Cancel the \(h\) terms:\[ \frac{-(3x^2 + 3xh + h^2)}{x^3 (x+h)^3} \]

Step 8: Final Simplified Expression

The final simplified expression is:\[ \frac{-(3x^2 + 3xh + h^2)}{x^3 (x+h)^3} \]

Key Concepts

Complex Fractions

A complex fraction is essentially a fraction within a fraction. It can seem intimidating at first glance. You might see fractions both in the numerator and in the denominator. But don't worry; simplifying these types of expressions can actually be straightforward. The key is to focus on simplifying the numerator and the denominator separately, before combining them into one final simplified expression.
For instance, in the given exercise, the numerator is itself a fraction: \( \frac{1}{(x+h)^3} - \frac{1}{x^3} \), which signifies two smaller fractions being subtracted from one another.

• First, identify the overall structure of the complex fraction.
• Find ways to simplify this by concentrating on the smaller fractions inside it.

Ultimately, the goal is to clean up the expression so that you have a single, manageable fraction to work with.

Finding Common Denominators

Finding a common denominator is often an essential step in simplifying rational expressions. It's especially important when dealing with more complex structures, like the fraction in the numerator of the exercise, \( \frac{1}{(x+h)^3} - \frac{1}{x^3} \).
To do this, identify the least common multiple (LCM) of the denominators involved; in this case, that would be \((x+h)^3 \cdot x^3\).
By multiplying each individual fraction by appropriate terms to obtain this common denominator, you can then combine them into a single, more easily manageable expression.

• Determine the LCM of all the present denominators.
• Rewrite each fraction involved using the determined LCM as the new denominator.

Once rewritten, you simplify further by combining the numerators.

Factorization

Factorization involves expressing a polynomial or expression as a product of its factors. This technique is particularly helpful in simplifying expressions further once you've dealt with their arithmetic complexities.
In the example solution, the expanded polynomial \(-3x^2h - 3xh^2 - h^3\) can be unfurled to express common factors. In this scenario, you can factor out \(h\) from the expression, helping you to simplify it further by canceling common factors with the denominator.

• Identify the greatest common factor in a polynomial expression.
• Factor it out to simplify further manipulations.

By doing this, you make the expression cleaner and easier to work with, often revealing cancel-out opportunities in complex fractions.

Polynomial Expansion

Polynomial expansion involves expanding expressions raised to a power. It's a crucial step in transforming complex expressions into simplified forms for easier manipulation.
Consider the expression \((x+h)^3\) from the exercise. By expanding this, you can express it as \(x^3 + 3x^2h + 3xh^2 + h^3\). Doing so allows you to work individual terms, simplifying the entire expression. This step prepares the polynomial for further factorizations or combinations in the solving process.

• Take each term in the binomial and apply the Binomial Theorem or distribute the powers.
• Break down each part to work with smaller, simpler components.

Once expanded, polynomials don't only simplify the given problem but also enable further factorization efforts, paving the way to reach the final simplified form.