Problem 44 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( 4x^3 + 6x^2h + 4xh^2 + h^3 \).

Step 1: Understand the Problem

We want to factor the expression \( \frac{(x+h)^{4}-x^{4}}{h} \). This expression represents the difference quotient of \( x^4 \) and using the Binomial Theorem helps express \( (x+h)^4 \) in terms of \( x \) and \( h \).

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \( (x+h)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} h^k \). Apply this to \( (x+h)^4 \) to get:\[(x+h)^4 = \binom{4}{0} x^4 + \binom{4}{1} x^3 h + \binom{4}{2} x^2 h^2 + \binom{4}{3} x h^3 + \binom{4}{4} h^4\]Simplifying, we have:\[(x+h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4\]

Step 3: Substitute into the Original Expression

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

Step 4: Simplify the Numerator

Notice that \(x^4\) cancels out:\[\frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}\]

Step 5: Factor \( h \) out of the Numerator

Factor \( h \) from each term in the numerator to get:\[h(4x^3 + 6x^2h + 4xh^2 + h^3)\]

Step 6: Cancel \( h \) in the Fraction

Cancel \( h \) from the numerator and denominator:\[4x^3 + 6x^2h + 4xh^2 + h^3\]

Step 7: Conclusion

The expression simplifies to \( 4x^3 + 6x^2h + 4xh^2 + h^3 \) after factoring using the Binomial Theorem.

Key Concepts

Difference Quotient

In calculus, you'll often encounter the concept of a difference quotient. It serves as the foundation for understanding derivatives, which measure how functions change. Think of the difference quotient as a way to quantify the rate of change of a function.

Mathematically, for a function \( f(x) \), the difference quotient is expressed as:\[\frac{f(x+h) - f(x)}{h}\]Here, \( h \) approaches zero. This formula essentially measures the average rate of change of the function as it moves from point \( x \) to \( x+h \).

• It's like calculating the slope of the secant line between two points on a curve.
• Critical for understanding tangent lines and instantaneous rates of change.

In our exercise, we observe this principle through the expression \( \frac{(x+h)^{4}-x^{4}}{h} \). This demonstrates the average rate of change of the polynomial \( x^4 \) as \( x \) shifts slightly by \( h \). Knowing this builds a strong base for deeper explorations in calculus, especially when learning to compute derivatives.

Factoring Algebraic Expressions

Factoring is a fundamental skill in algebra, where we rewrite expressions as a product of simpler factors. For our given expression, we focused on factoring out common terms in the numerator before simplifying.

• Factoring makes equations easier to solve, by breaking them into products of simpler expressions.
• Helps in simplifying expressions and solving polynomial equations.

In our example, after applying the Binomial Theorem, the expression becomes \( 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \). We observed \( h \) as a common factor in each term of the numerator. By factoring \( h \), the expression simplifies to \( h(4x^3 + 6x^2h + 4xh^2 + h^3) \).
This step is crucial as it prepares the expression for further simplification, where \( h \) is eventually canceled from the numerator and the denominator. Mastering factoring is invaluable for managing complex algebraic expressions.

Polynomial Expressions

Polynomials are mathematical expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomial expressions is vital as they frequently appear in algebraic problems and beyond.
Polynomials can be:

• Simple, like \( x+2 \) or complex, as we see with \( (x+h)^4 \).
• Analyzed for roots, factored, and expanded using various algebraic properties.

In our exercise, expanding \( (x+h)^4 \) using the Binomial Theorem resulted in a complex polynomial expression with multiple terms. This decomposition made it easier to analyze each part separately.
Within the expression \( 4x^3 + 6x^2h + 4xh^2 + h^3 \), understanding the structure helps in differentiating and integrating polynomial forms, essential skills in calculus. Recognizing patterns in polynomials enhances problem-solving efficiency and sharpens analytical skills in mathematics.