Problem 42 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The 12th term is \(139776x^5 y^{22}\).

Step 1: Understand Binomial Theorem

The binomial theorem is used to expand expressions of the form \((a+b)^n\). According to the theorem, \((a+b)^n\) can be expanded as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \).

Step 2: Identify the Parameters

In the expression \((2x + y^2)^{16}\), identify \(a = 2x\), \(b = y^2\), and \(n = 16\). We are looking for the 12th term.

Step 3: Determine the Term Number

For finding the 12th term in the expansion, recall that the (k+1)-th term from a binomial expansion \((a+b)^n\) is given by \(\binom{n}{k} a^{n-k} b^k\). Thus, for the 12th term, set \(k+1 = 12\), which gives \(k = 11\).

Step 4: Apply Binomial Formula for the 12th Term

Using \(k=11\), the term can be calculated as: \[ \binom{16}{11} (2x)^{16-11} (y^2)^{11} \].

Step 5: Calculate the Coefficient and simplify

Compute the binomial coefficient \( \binom{16}{11} = 4368 \). Then calculate \((2x)^5 = 32x^5\) and \((y^2)^{11} = y^{22}\). Multiply these terms to get: \[ 4368 \times 32x^5 \times y^{22} = 139776x^5 y^{22} \].

Key Concepts

Binomial Expansion

A binomial expansion is a method for expressing the power of a binomial as a sum. This concept stems from the Binomial Theorem, which states that any expression in the form \((a + b)^n\) can be expanded into a sum involving terms of the form \(a^{n-k}b^k\). Each term in the expansion has a binomial coefficient, denoted as \(\binom{n}{k}\).

• The Binomial Theorem allows us to find any term in the expansion of a binomial expression without having to multiply the expression repeatedly.
• This theorem is particularly useful for large values of \(n\), as it provides a formulaic approach to expansion.

By applying the theorem, you can quickly determine the expression \((2x + y^2)^{16}\) by identifying the individual terms that compose the expansion. Each term will increase in power for \(b\) while decreasing for \(a\), providing a structured and predictable array of terms.

Binomial Coefficient

The binomial coefficient plays a crucial role in binomial expansions. It is represented by the symbol \(\binom{n}{k}\), which is read as "n choose k." This coefficient determines the number of ways to choose \(k\) items from \(n\) items without regard to order.

• To compute \(\binom{n}{k}\), use the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• For instance, in the expression \(\left(2x + y^2\right)^{16}\), if we need the 12th term, we calculate \(\binom{16}{11}\), resulting in 4368.

This coefficient tells you how many times each term should appear when expanding the binomial. Furthermore, it helps in understanding the distribution of terms across the expansion, which is crucial in complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. They play an integral role in various branches of mathematics and serve as the foundation for algebra.

• In the context of the binomial theorem, algebraic expressions like \((2x + y^2)\) are raised to powers, significantly increasing their complexity.
• Breaking down such expressions into expanded forms simplifies operations and provides insights into their structure.

Consider the expression \((2x + y^2)^{16}\); its expanded form provides a clear picture of each individual term's role. Use these expansions to grasp complex polynomial behaviors, underlying relationships, and patterns in mathematics, promoting deeper mathematical insights and applications.