Problem 53 - Algebra and Trigonometry

Question

Find\(\frac{f(x+h)-f(x)}{h}\) and simplify. $$ f(x)=x^{4}+7 $$

Step-by-Step Solution

Answer

Therefore, the simplified form of the difference quotient of the function \(f(x)=x^{4}+7\) is \(4x^{3} + 6x^{2}h + 4xh^{2} + h^{3}\)

Step 1: Substitute the function

Firstly, the function \(f(x)=x^{4}+7\) is substituted into the formula \(\frac{f(x+h)-f(x)}{h}\). The result is: \(\frac{((x+h)^{4} + 7) - (x^{4}+7)}{h}\)

Step 2: Simplify the expression

Next up, simplify the expression through expansion and also through subtracting similar terms. When expanded, you get: \(\frac{x^{4} + 4x^{3}h + 6x^{2}h^{2} + 4xh^{3} + h^{4} - x^{4}}{h}\). After simplifying, you are left with \(\frac{4x^{3}h + 6x^{2}h^{2} + 4xh^{3} + h^{4}}{h}\)

Step 3: Cancel out h

Sharing letters i.e. cancelling out \(h\) leaves: \(4x^{3} + 6x^{2}h + 4xh^{2}+ h^{3}\)

Step 4: Simplify the expression

We simplify this expression furthermore and our final expression becomes \(4x^{3} + 6x^{2}h + 4xh^{2} + h^{3}\)

Key Concepts

Function Substitution

In mathematics, function substitution is a technique where we replace a variable in a formula with the actual expression that defines the function. This allows us to manipulate and work with the functions directly. In the given exercise, we start with the function \(f(x) = x^4 + 7\). The key here is to substitute \(f(x+h)\) into the expression \(\frac{f(x+h) - f(x)}{h}\). This involves substituting \(x+h\) into the function, which results in \((x+h)^4 + 7\) replacing \(f(x+h)\). Thus, the expression becomes:

• The numerator is \((x+h)^4 + 7 - (x^4 + 7)\)
• This gives us \((x+h)^4 - x^4\) when the constants \(+7\) cancel each other out.

By successfully substituting the function, we are set up to further simplify the formula.

Polynomial Expansion

Polynomial expansion, or expanding a polynomial, refers to the process of expanding an expression that is raised to a power. This is applicable when substituting \(x+h\) into the function. In our expression \((x+h)^4\), we expand it using the binomial theorem or distribution. It involves calculating and adding up each term from expanding the bracket:

• Start with \((x+h)^4\), which involves terms like:
• \(x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4\)
Each term arises from distributing the exponent over the binomial.
• Notice \(x^4\) cancels out with \(-x^4\) from the original function.

This leaves us with the polynomial \(4x^3h + 6x^2h^2 + 4xh^3 + h^4\). Expanding the polynomial correctly sets us up to manipulate the expression further.

Expression Simplification

Expression simplification is a fundamental skill in mathematics, especially when working with algebraic fractions. After expanding the polynomial, it's time to simplify the expression further. For the difference quotient, simplification involves reducing the expanded polynomial:

• Initially, we have \(\frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}\)
• To simplify this, we check if there is a common factor. Here, each term in the numerator has an \(h\).
• Cancelling \(h\) from the numerator and denominator leads to the simplified expression:
• \(4x^3 + 6x^2h + 4xh^2 + h^3\)

As terms are simplified, expressions are easier to understand and work with.

Canceling Terms

Canceling terms in algebra involves removing equal values or expressions from the numerator and the denominator. It's a process that simplifies fractions and reveals simpler forms of expressions:

• In our fraction \(\frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}\), each term in the numerator has the factor \(h\).
• Dividing both the numerator and denominator by \(h\) effectively cancels it out.
• This gives \(4x^3 + 6x^2h + 4xh^2 + h^3\), a much simpler polynomial without the fraction form.

Canceling terms streamlines the complexity of expressions and completes the simplification, making it easier to handle further mathematical operations.