Problem 26 - Precalculus: Mathematical for Calculus

Question

Find the first four terms in the expansion of \(\left(x^{1 / 2}+1\right)^{30}\)

Step-by-Step Solution

Answer

The first four terms are: \(x^{15}, 30x^{14.5}, 435x^{14}, 4060x^{13.5}\).

Step 1: Recognize the binomial theorem

The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this problem, we can set \(a = x^{1/2}\), \(b = 1\), and \(n = 30\). The goal is to find the first four terms in the expansion.

Step 2: Calculate the first term

The first term corresponds to \(k = 0\). Using the binomial formula, the first term is \( \binom{30}{0} (x^{1/2})^{30} (1)^0 = 1 \cdot x^{15} \cdot 1 = x^{15} \).

Step 3: Calculate the second term

The second term corresponds to \(k = 1\). The second term is calculated as follows: \( \binom{30}{1} (x^{1/2})^{29} (1)^1 = 30 \cdot x^{14.5} \cdot 1 = 30x^{14.5} \).

Step 4: Calculate the third term

The third term corresponds to \(k = 2\). Using the binomial formula, the third term is \( \binom{30}{2} (x^{1/2})^{28} (1)^2 = 435 \cdot x^{14} \cdot 1 = 435x^{14} \).

Step 5: Calculate the fourth term

The fourth term corresponds to \(k = 3\). This term is \( \binom{30}{3} (x^{1/2})^{27} (1)^3 = 4060 \cdot x^{13.5} \cdot 1 = 4060x^{13.5} \).

Key Concepts

Exponential Expressions

When dealing with algebraic expressions, you often encounter exponents, which are used to represent repeated multiplication. An exponential expression comprises a base and an exponent. For example, in the expression \(x^{1/2}\), \(x\) is the base, and \(1/2\) is the exponent, meaning the square root of \(x\).
An exponential expression with a fractional exponent can be rewritten in radical form. More crucially, it allows us to work with calculations in various algebraic techniques efficiently.
These expressions are powerful in mathematics as they can significantly simplify multiplication and division of numbers or variables. They are also integral in simplifying polynomial expressions, which is precisely what the binomial theorem exploits.

Binomial Expansion

The binomial expansion is a crucial algebraic technique that expresses a binomial raised to any power as a sum of terms. This is governed by the Binomial Theorem, which states:
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
This theorem helps break down complex expressions into simpler forms. For the binomial \((x^{1/2} + 1)^{30}\), "\(a\)" is \(x^{1/2}\), and "\(b\)" is 1. Each term in the expansion represents a combination of the two parts of the binomial raised to respective powers that sum to \(n\).
The coefficients of these terms \(\binom{n}{k}\) follow directly from combinations, highlighting the relationship between algebra and combinatorics. Thus, the binomial expansion is a tool that transforms exponential expressions into manageable polynomial forms.

Combinatorics

Combinatorics is the study of counting, and it plays a critical role in binomial expansions. It deals with the selection of items from a collection, where the order may or may not matter.
In the context of the binomial expansion \((x^{1/2} + 1)^{30}\), combinatorics determines the binomial coefficients \(\binom{30}{k}\).
These coefficients represent the number of ways to choose \(k\) elements from a set of \(30\) elements, underscoring the strategic selection involved.
Each term in the binomial expansion is weighted by these coefficients, which reflects their combinatorial roots. As such, understanding combinatorics opens up new approaches to solving problems involving polynomials and exponential expressions, and it closes the loop between abstract mathematical principles and real-world counting applications.