Problem 48 - Precalculus: Mathematics for Calculus

Question

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 Problem

We need to simplify the expression \( \frac{(x+h)^{4}-x^{4}}{h} \). It involves expanding \((x + h)^4\) using the Binomial Theorem and simplifying.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, \(a = x\), \(b = h\), and \(n = 4\). We expand \((x + h)^4\): \((x + h)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} h^k = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \).

Step 3: Substitute the Binomial Expansion

Substitute the expansion of \((x + h)^4\) back into the original expression: \(\frac{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h} \).

Step 4: Simplify the Expression

Cancel \(x^4\) in the numerator to get \(\frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h} \).

Step 5: Factor Out \(h\)

Factor \(h\) out of each term in the numerator: \(\frac{h(4x^3 + 6x^2h + 4xh^2 + h^3)}{h} \).

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

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

Key Concepts

Polynomial Expansion

The concept of polynomial expansion is crucial when dealing with expressions like \((x + h)^4\). In mathematics, the Binomial Theorem offers a systematic approach to expanding binomials. A binomial is simply an expression containing two terms, like \((x + h)\). The theorem provides a formula to expand expressions of the form \((a + b)^n\), where \(n\) is a positive integer. The formula is: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, \(\binom{n}{k}\) is a binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\). This coefficient determines how many ways \(k\) selections can be made from \(n\) items.For the expression \((x + h)^4\), you set \(a = x\), \(b = h\), and \(n = 4\). Expanding this using the Binomial Theorem, you get:- \(x^4\) (when \(k=0\))- \(4x^3h\) (when \(k=1\))- \(6x^2h^2\) (when \(k=2\))- \(4xh^3\) (when \(k=3\))- \(h^4\) (when \(k=4\))Putting them together, \((x + h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4\). This systematic expansion makes solving binomial expressions manageable and paves the way for simplifying complex algebraic expressions.

Simplification

Simplification is a process to make an expression easier to understand and solve by reducing its complexity. In simplifying \(\frac{(x+h)^{4}-x^{4}}{h}\), the first step is expanding \((x + h)^4\) using the Binomial Theorem. After expanding, the expression inside the numerator transforms to:\[x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4\]To simplify further, \(x^4\) can be cancelled out, leaving:\[4x^3h + 6x^2h^2 + 4xh^3 + h^4\]This expression is still over \(h\), so the next step in simplification requires factoring. Factoring \(h\) out of each term in the numerator simplifies the problem significantly:\[\frac{h(4x^3 + 6x^2h + 4xh^2 + h^3)}{h}\]Notice how each term in the numerator contains \(h\). This common factor allows canceling \(h\) in both the numerator and the denominator, leaving us with:\[4x^3 + 6x^2h + 4xh^2 + h^3\]This reduced expression is simpler and much easier to handle. Simplification like this makes complex algebraic expressions more accessible and manageable.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They are the building blocks of algebra. Understanding how to manipulate these expressions is essential in solving mathematical problems. Consider the expression we are simplifying: \(\frac{(x+h)^4-x^4}{h}\).Algebraic expressions can include terms, which are parts separated by a plus or minus sign. In this exercise, terms like \((x+h)^4\) and \(x^4\) are combined in a fraction, with operations like subtraction and division involved.When handling algebraic expressions, one must:

• Identify and apply mathematical operations, like addition, subtraction, multiplication, and division.
• Use algebraic rules, such as factoring and distributing.
• Apply theorems or formulas as needed, such as the Binomial Theorem for expansion.
• Simplify the expression by canceling or combining like terms.

In this exercise, the manipulation of algebraic expressions involves skills like expansion using the Binomial Theorem, subtraction, factoring out common variables, and simplifying fractions. These processes illustrate the broad utility and flexibility of algebra in breaking down complex mathematical problems.