Problem 25 - Precalculus: Mathematical for Calculus

Question

Find the first three terms in the expansion of \((x+2 y)^{20}\)

Step-by-Step Solution

Answer

The first three terms are \(x^{20}, 40x^{19}y, 760x^{18}y^2\).

Step 1: Identify the Binomial Theorem

To find specific terms in the expansion of \((x+2y)^{20}\), we use the binomial theorem. The theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this problem, \(a = x\), \(b = 2y\), and \(n = 20\).

Step 2: Find the First Term

The first term corresponds to \(k = 0\). Substitute into the binomial formula: \(\binom{20}{0} x^{20-0} (2y)^0 = 1 \cdot x^{20} \cdot 1 = x^{20}\).

Step 3: Find the Second Term

The second term corresponds to \(k = 1\). Substitute into the binomial formula: \(\binom{20}{1} x^{20-1} (2y)^1 = 20 \cdot x^{19} \cdot 2y = 40x^{19}y\).

Step 4: Find the Third Term

The third term corresponds to \(k = 2\). Substitute into the binomial formula: \(\binom{20}{2} x^{20-2} (2y)^2 = 190 \cdot x^{18} \cdot 4y^2 = 760x^{18}y^2\).

Step 5: Compile the First Three Terms

Compile the terms identified in Steps 2, 3, and 4. The first three terms of \((x+2y)^{20}\) are \(x^{20} + 40x^{19}y + 760x^{18}y^2\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a fundamental concept in algebra. It involves expressing a power of a polynomial sum as a series of terms. Each term in the expansion consists of products of the polynomial's parts raised to varying powers.

One way to expand a polynomial raised to a power, like \((x+2y)^{20}\), is by using **expansion techniques** such as the Binomial Theorem. This technique converts the expression into a sum of terms that can be more easily handled and understood.

• The polynomial is expanded into individual terms each involving powers of different parts of the original expression.
• Each term contributes to the overall expansion and becomes part of the final polynomial form.

Understanding polynomial expansion is essential for simplifying complex algebraic expressions and is used across mathematics in calculus, statistics, and beyond.

Binomial Coefficients

Binomial coefficients are central to polynomial expansion through the Binomial Theorem. They give the **number of ways** to choose a subset of elements from a larger set. In mathematical terms, a binomial coefficient is written as \( \binom{n}{k} \), read as "n choose k."

For example, in the expression \((x+2y)^{20}\), the coefficient for any term is computed using:

• \( \binom{20}{0} \) for the first term, leading to \(x^{20}\).
• \( \binom{20}{1} \) for the second term, providing coefficients that expand it into \(40x^{19}y\).
• \( \binom{20}{2} \) for the third term, used to determine \(760x^{18}y^2\).

Binomial coefficients are derived from **combinatorial principles** and are essential for calculating terms in polynomial expansions. They can be easily found using Pascal's Triangle or calculated directly using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(!\) denotes factorial.

Mathematical Proofs

Mathematical proofs in the context of the Binomial Theorem provide the **logical foundation** that assures us of the validity of the theorem itself. They help ensure that formulas like those in polynomial expansions are not just coincidental but are universally true.

To prove the Binomial Theorem, mathematicians use **induction** or combinatorial arguments:

• **Inductive Proof:** This method involves proving that if the theorem holds for one integer, it holds for the next. This involves a base case, where if proven true for, say, \( n = 1 \), it logically extends to \( n = 2, 3, \)...
• **Combinatorial Proof:** This involves reasoning based on counting methods, explaining how the different combinations of expanded terms arise.

Understanding proofs enriches one's ability to see the structure behind mathematical truths, leading to deeper insights and appreciation for the order within algebraic concepts like polynomial expansion.