Problem 44 - Algebra and Trigonometry

Question

\(43-44\) Simplify using the Binomial Theorem. $$ \frac{(x+h)^{4}-x^{4}}{h} $$

Step-by-Step Solution

Answer

The simplified expression is \(4x^3 + 6x^2h + 4xh^2 + h^3\).

Step 1: Understand the Expression

The given expression is \( \frac{(x+h)^{4}-x^{4}}{h} \). It involves simplifying a difference quotient using the Binomial Theorem.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \((x + h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4\). Substitute this expansion into the expression.

Step 3: Substitute the Expansion

Replace \((x+h)^4\) in the original expression: \( \frac{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h} \).

Step 4: Simplify the Numerator

Cancel like terms in the numerator: \(x^4 - x^4\) cancels to zero, leaving \(4x^3h + 6x^2h^2 + 4xh^3 + h^4\).

Step 5: Factor the Remaining Terms

Factor out \(h\) from the remaining terms: \(h(4x^3 + 6x^2h + 4xh^2 + h^3)\).

Step 6: Simplify the Expression

Cancel \(h\) in the numerator and denominator: \(4x^3 + 6x^2h + 4xh^2 + h^3\).

Step 7: Final Expression

The simplified form of the expression is \(4x^3 + 6x^2h + 4xh^2 + h^3\).

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus. It is essential for understanding how functions change, serving as the foundation for derivatives. In the expression given, the difference quotient is represented as \( \frac{(x+h)^{4}-x^{4}}{h} \).
This setup reflects the change in a function's values over a small interval \( h \).

• The numerator \((x+h)^4 - x^4\) shows the difference in the function value due to a small change \( h \).
• The denominator \( h \) represents the size of that change.

By simplifying this expression, we can gain insight into how the function \((x)^4\) behaves as \( x \) changes. This process is crucial for finding derivatives, which in turn helps understand rates of change.
When functions involve higher powers, like \( (x+h)^4 \), methods like the Binomial Theorem help in expanding and simplifying the expression.

Algebraic Simplification

Algebraic simplification in expressions is key to making complex problems easier to handle. Once \( (x+h)^4 \) is expanded using the Binomial Theorem, the next step is to simplify the resulting expression. This involves canceling terms and reducing complicated parts into simpler forms.
For the expression \( \frac{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h} \), note the following:

• The \( x^4 \) term in the expansion cancels with \( -x^4 \), removing them from the equation.
• What remains is \( 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \), all terms now involving \( h \).

This process reduces the original expression step by step, making it straightforward to analyze and further simplify. By factoring out \( h \), we achieve a form that is easier to work with.

Polynomial Expansion

Polynomial expansion is a powerful technique for dealing with expressions like \( (x+h)^4 \). The Binomial Theorem allows us to break down such expressions into a series of terms that are more manageable. The theorem states that any binomial \((a + b)^n\) can be expanded using the formula:
\[\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
In our case:

• \((x + h)^4\) becomes \(x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \).

Each term represents a combination of powers from \(x\) and \(h\), based on the coefficients determined from the binomial coefficients. This expansion makes it possible to explore and simplify complex algebraic expressions that involve variables raised to higher powers. Through expanding and simplifying, we can better understand and manipulate polynomial relationships.