Problem 14 - Algebra 2 and Trigonometry

Question

In \(12-17,\) write the \(n\) th term of each binomial expansion. $$ (2 x+y)^{6}, n=4 $$

Step-by-Step Solution

Answer

The 4th term in the expansion is \(160x^3y^3\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem is used to expand expressions of the form \((a+b)^n\). It states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).

Step 2: Identify the Parameters

In the given expression \((2x + y)^6\), we identify \(a = 2x\), \(b = y\), and \(n = 6\). For the 4th term, we will use \(k = 3\) as indices start from zero; hence, the 4th term corresponds to \(k = 3\).

Step 3: Calculate the Binomial Coefficient

The binomial coefficient for the 4th term is \(\binom{6}{3}\). Calculate it using \(\binom{6}{3} = \frac{6!}{3! (6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\).

Step 4: Determine the Powers of Each Term

For the 4th term \(T_{4}\), calculate the powers of \(a = 2x\) and \(b = y\) as follows: \((2x)^{6-3} = (2x)^3\) and \(y^3\).

Step 5: Evaluate the Expression

Substitute these calculated values into the term: \[T_{4} = \binom{6}{3} \cdot (2x)^3 \cdot y^3 = 20 \cdot 8x^3 \cdot y^3 = 160x^3y^3\].

Step 6: State the nth Term

The nth term of the binomial expansion for \((2x + y)^6\) when \(n=4\) is \(160x^3y^3\).

Key Concepts

The Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra that simplifies the expansion of expressions raised to a power. It allows you to easily expand expressions of the form \((a + b)^n\). The theorem states:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

This formula gives a way to break down the expansion into a series of terms, where each term is a product of powers of \(a\) and \(b\) multiplied by a binomial coefficient. The binomial coefficients are crucial here and are determined by the structure of the problem.
This method provides a systematic approach, incredibly useful for higher powers where direct multiplication becomes impractical.

Understanding the Binomial Coefficient

The binomial coefficient \(\binom{n}{k}\) is an essential part of the Binomial Theorem. It determines the number of ways to choose \(k\) elements from \(n\) elements, which corresponds to the coefficients in the binomial expansion.

• The formula for the binomial coefficient is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\). For example, to find the coefficient for the 4th term in the expansion of \((2x + y)^6\), we calculate \(\binom{6}{3} = 20\).
This coefficient is then used to determine the weight of each term in the expansion. Understanding this concept is crucial for correctly applying the theorem to find individual terms.

Polynomial Expansion with Binomials

Polynomial expansion involves expressing a power of a binomial as a sum of terms. Each term in this sum consists of products of powers of the binomial's constituents. When you expand \((2x + y)^6\), you are essentially distributing the powers across all possible products of \(2x\) and \(y\).

• For example, the 4th term requires calculating both individual powers and the coefficient: \((2x)^3\) and \(y^3\), along with \(\binom{6}{3}\) as the coefficient.

In this specific instance, the sub-expression evaluates to \(160x^3y^3\). The process is methodical, relying on the combination of powers and coefficients.

Pascal's Triangle Connection

Pascal's Triangle is a valuable tool for easily determining binomial coefficients without direct computation. Each level of this triangular arrangement of numbers represents the coefficients for corresponding powers of binomial expansions.

• The row numbers correspond to the power \(n\) in the binomial expression, while each position in the row aligns with different \(k\) values, representing terms in the expansion.

For example, the 6th row in Pascal's Triangle is \(1, 6, 15, 20, 15, 6, 1\), which matches the coefficients for the expansion of \((a + b)^6\). By using Pascal's Triangle, you can instantly find any binomial coefficient needed, providing a quick visual aid during calculations.