Problem 39 - Algebra and Trigonometry

Question

Find the term indicated in each expansion. $$ (2 x+y)^{6} ; \text { third term } $$

Step-by-Step Solution

Answer

The third term in the expansion \((2x+y)^6\) is \(240x^4y^2\).

Step 1: Determine the Values

From the problem, we identify that \(a = 2x\), \(b = y\), \(n = 6\) and the term number \(r = 2\) (because \(r\) starts from 0, so the third term will have \(r = 2\)).

Step 2: Apply the binomial theorem formula

We'll substitute our values into the binomial theorem formula. Remember the formula is: \[ T_{r+1} = ^nC_r * a^{n-r} * b^r \] So we get: \[ T_{3} = ^6C_2 * (2x)^{6-2} * y^2 \]

Step 3: Calculate the combination

To calculate \(^6C_2\), we can use the formula for combinations: \[ ^6C_2 = \frac{6!}{2!(6-2)!} \] which simplifies to \(15\).

Step 4: Simplify the expression

With the combination calculated, we can simplify the equation from step 2 to get the third term: \[ T_{3} = 15 * (2x)^4 * y^2 \] Simplify further to: \[ T_{3} = 240x^4y^2 \]

Key Concepts

Expansion

In mathematics, the concept of **expansion** refers to expressing a mathematical expression as a sum of terms. When working with the binomial theorem, expansion involves expressing the power of a binomial, like \((a + b)^n\), as a sum of multiple terms. Each term is unique based on its coefficients and the powers of the variables involved, such as \((2x + y)^6\).
The binomial expansion allows us to break down complex expressions into simpler ones. This is useful because it enables us to find specific terms without needing to compute the entire expression right away. We use the binomial formula to determine the coefficients and powers of each term. It makes complex problems more accessible by revealing patterns that emerge in binomial expansions, such as consistent symmetries and coefficient distributions.
Ultimately, learning how to expand binomials using the binomial theorem makes it much easier to solve equations and extract valuable insights, particularly in problems relating to probability and algebra.

Combinations

Combinations are a fundamental concept in combinatorics, a branch of mathematics studying how objects can be arranged or selected. In the context of the binomial theorem, **combinations** help determine the coefficients of each term in the expansion. When given \((a + b)^n\), the number of ways to select particular arrangements of terms (like how many terms should include \(a\) or \(b\)) is given by the formula of combinations:\[^nC_r = \frac{n!}{r!(n-r)!}\]

Understanding Combinations

- **Symbol Usage:** The notation \(^nC_r\) (read as "n choose r") denotes the number of combinations.- **Factorials:** The use of factorials \((n!)\) represents the product of an integer and all positive integers below it.
The role of combinations in the binomial theorem is significant because they determine how terms are formed in the binomial expansion. For example, in our exercise where \((2x + y)^6,\) the exercise uses \(^6C_2\) to find the coefficient of the third term, which maps directly to factors of the expanded terms.

Binomial Coefficients

At the heart of the binomial theorem lies the concept of **binomial coefficients**. These coefficients are the numerical factors associated with each term in the expansion of a binomial expression. For the binomial expression \((a + b)^n\), each term can be represented as \(^nC_r\), where \(^nC_r\) is the binomial coefficient. This helps us determine the multiplier for each term in the expansion process.

Calculation of Binomial Coefficients

Each binomial coefficient is calculated using the combination formula:- For example, the calculation \(^6C_2\) in our exercise results in 15. This tells us that there are 15 combinations of choosing 2 terms from 6, which guides us to our term's exact coefficient.

Significance in Expansion

Binomial coefficients ensure that as you progress through each term in the binomial expansion, the coefficients correctly reflect the number of ways terms can be formed. They are symmetrical, and each of them represents a specific distribution of a's and b's in the expanded terms. This pattern is famously illustrated by Pascal's triangle and shows consistently in binomial expansions.