Problem 42 - A Graphical Approach to College Algebra

Question

Write the indicated term of each binomial expansion. Twelfth term of \(\left(2 x+y^{2}\right)^{16}\).

Step-by-Step Solution

Answer

139776x^5y^{22}

Step 1: Identify the general term in the binomial expansion

The general term in the expansion of \((a+b)^n\) is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). Here, \(a = 2x\), \(b = y^2\), and \(n = 16\).

Step 2: Determine the term number (k)

We want the twelfth term of the expansion. The term \(T_{k+1}\) corresponds to \(k = 11\) for the twelfth term. So, we use \(k = 11\) in the formula for the general term.

Step 3: Calculate the binomial coefficient

The binomial coefficient is \(\binom{16}{11}\). Calculate it using the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]. So \[\binom{16}{11} = \frac{16!}{11!5!} = 4368\].

Step 4: Compute the powers of a and b

Substitute \(k = 11\) into \(a^{n-k}\) and \(b^k\). We have \((2x)^{16-11} = (2x)^5 = 32x^5\) and \((y^2)^{11} = y^{22}\).

Step 5: Compute the twelfth term

Combine the results from previous steps: \[T_{12} = \binom{16}{11} a^{16-11} b^{11} = 4368 \cdot 32x^5 \cdot y^{22} = 139776x^5y^{22}\].

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful mathematical tool that helps expand expressions raised to a power, such as \( (a + b)^n \). It provides a formula for finding any term in the expansion without having to multiply the expression out completely. Consider:

• The formula for the \(k+1\)th term in the expansion, also known as the general term, is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\).
• This breaks the expression into easily calculable parts using combinations and powers.

This theorem is widely used because it simplifies the process of binomial expansion, making complex expressions more manageable.

Pascal's Triangle

Pascal's Triangle is a geometric arrangement of numbers, where each number is the sum of the two numbers directly above it. This triangle provides a quick way to find the coefficients needed for binomial expansion.

• The rows in Pascal's Triangle correspond to the powers in a binomial expansion.
• For example, the coefficients of \( (a+b)^4 \) can be found in the 5th row of the triangle: \(1, 4, 6, 4, 1\).

Using Pascal’s Triangle can save time and effort, as it eliminates the need for calculations using factorials.

Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the terms of a binomial expansion. They are represented by \({n \choose k}\) and can be calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

• The symbol \(n!\) represents the factorial of \(n\), which is the product of all positive integers up to \(n\).
• These coefficients are symmetrical; this means \(\binom{n}{k} = \binom{n}{n-k}\).
• In our specific example, \(\binom{16}{11} = 4368\), calculated using the factorial method.

Understanding how to calculate these coefficients is crucial in applying the binomial theorem effectively.