Problem 49 - College Algebra

Question

If \(f(x)=x^{4},\) find \(\frac{f(x+h)-f(x)}{h}\) and simplify.

Step-by-Step Solution

Answer

The simplified result of the difference quotient applied to the function \(f(x)=x^{4}\) is \(4x^{3}+6x^{2}h+4xh^{2}+h^{3}\).

Step 1: Plug into Difference Quotient Formula

Substitute the function, \(f(x)=x^{4}\), into the difference quotient, giving \(\frac{(x+h)^{4}-x^{4}}{h}\).

Step 2: Expand the Binomial

Expand the binomial, \((x+h)^{4}\), which results in \(x^{4}+4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}\).

Step 3: Simplify the Expression

Simplify \(\frac{(x+h)^{4}-x^{4}}{h}\) by substituting \(x^{4}+4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}\) for \((x+h)^{4}\), which yields \(\frac{x^{4}+4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}-x^{4}}{h}\). The \(x^{4}\) terms cancel out, leaving \(\frac{4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}}{h}\).

Step 4: Divide Each Term by h

Cancel out the h in every term, yielding \(4x^{3}+6x^{2}h+4xh^{2}+h^{3}\).

Step 5: Review the Result

The simplified form of the difference quotient applied to the function \(x^{4}\) is \(4x^{3}+6x^{2}h+4xh^{2}+h^{3}\).

Key Concepts

Binomial Expansion

Binomial expansion is a mathematical technique that involves expanding expressions that are raised to a power, typically written as \( (a + b)^n \). This expansion is governed by the Binomial Theorem, which allows you to express the expansion as a sum of terms involving binomial coefficients.

For example, in the exercise provided, \( (x + h)^4 \) is a binomial raised to the fourth power. When expanding this expression, we use the binomial coefficients that correspond to the expansion of a binomial raised to the fourth power. These coefficients are found in Pascal's Triangle or can be calculated using the formula for combinations: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(!\) denotes factorial.

The expansion of the given binomial, in this case, results in \( x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \), where the coefficients (1, 4, 6, 4, 1) reflect the fourth row of Pascal's Triangle. Understanding this method is crucial for efficiently dealing with expressions as seen in the exercise.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is often about making them more manageable and easier to interpret. This process involves several steps, including expanding expressions like binomials, combining like terms, and reducing fractions.

In the context of the exercise, simplification begins after expanding the binomial. The next step is to cancel like terms, as seen with the \( x^4 \) terms in the provided solution. This is followed by dividing terms by \( h \), which is a common factor. Once you eliminate the common factor, the expression \( f(x+h)-f(x))/h \) becomes \( 4x^3 + 6x^2h + 4xh^2 + h^3 \), where each term is free from the variable \( h \) in the denominator. It's important to ensure that the terms are combined correctly and that all like terms are consolidated to achieve the simplest form.

Rational Expressions

Rational expressions are fractions that contain polynomials in their numerator and denominator. Simplifying rational expressions entails factoring polynomials, canceling common factors, and making sure to state any restrictions on the variables that would make the denominator equal to zero.

In our difference quotient exercise, the original expression \( \frac{(x+h)^4 - x^4}{h} \) is a rational expression. The key step in simplifying such an expression is to eliminate the common factor that appears in all terms of the numerator, which in this case is \( h \). After the binomial expansion and cancelation of \( x^4 \), we are left with \( h \) in each term of the numerator. Once we divide each of these terms by \( h \), the expression is no longer a fraction, technically speaking, although it arose from a rational expression. Simplification of rational expressions is a fundamental skill in algebra that helps in solving equations, graphing functions, and performing calculus operations like finding the difference quotient.