Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (a+b)^{17} $$

Step-by-Step Solution

Answer

The first three terms are \(a^{17}\), \(17a^{16}b\), and \(136a^{15}b^2\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). Here, \(n\) is 17, which means we want to expand \((a+b)^{17}\) and find the first three terms.

Step 2: Calculate the First Term

The first term of the binomial expansion corresponds to \(k=0\). It is given by \(\binom{17}{0} a^{17-0} b^0 = 1 \cdot a^{17} \cdot 1 = a^{17}\). Thus, the first term is \(a^{17}\).

Step 3: Calculate the Second Term

The second term corresponds to \(k=1\). Calculate it as \(\binom{17}{1} a^{17-1} b^1 = 17 \cdot a^{16} \cdot b = 17a^{16}b\). So, the second term is \(17a^{16}b\).

Step 4: Calculate the Third Term

The third term corresponds to \(k=2\). Calculate it as \(\binom{17}{2} a^{17-2} b^2 = \frac{17 \cdot 16}{2!} a^{15} b^2 = 136a^{15}b^2\). Therefore, the third term is \(136a^{15}b^2\).

Key Concepts

Binomial Expansion

Binomial expansion is a fundamental concept in algebra that allows us to expand expressions that are raised to a power. It is particularly useful in dealing with expressions of the form \((a + b)^n\). According to the Binomial Theorem, such an expression can be written as the sum of terms: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\] This equation implies that when you expand \((a + b)^n\), each term in the expansion is formed by combining powers of \(a\) and \(b\) in a specific way. Here are the key points of binomial expansion:

• The power \(n\) determines the number of terms in the expansion, which is \(n+1\) terms in total.
• The coefficients of the terms are binomial coefficients, represented by \(\binom{n}{k}\).
• Each term has the form \(\binom{n}{k} a^{n-k} b^k\) where \(k\) ranges from \(0\) to \(n\).

The application of this theorem enables us to expand and solve binomial expressions systematically, making it possible to calculate specific terms without needing to expand the entire expression.

Pascal's Triangle

Pascal's Triangle is a fascinating mathematical tool that helps us easily find binomial coefficients, which are essential for binomial expansion. Imagine it as a triangular array of numbers arranged in rows, where each number is the sum of the two numbers directly above it in the previous row. This arrangement starts with a single 1 at the top. The rows of Pascal’s Triangle correspond to the coefficients you'd find in the expansion of \((a+b)^n\).

• The first row \((n=0)\) is just \([1]\), and the second row \((n=1)\) is \([1, 1]\).
• The third row \((n=2)\) has \([1, 2, 1]\), which are the coefficients for \((a + b)^2\).
• This pattern continues on, growing with each additional row.

With Pascal's Triangle, calculating binomial coefficients like \(\binom{n}{k}\) becomes an easy task as you just have to locate the right row and position. This straightforward method saves time and helps to quickly verify our expansion calculations by providing a visual reference for binomial coefficients.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, arrangement, and combination of elements in sets. It underpins the calculation of binomial coefficients in binomial expansion. When we see \(\binom{n}{k}\), it's known as a "binomial coefficient," which is pivotal in combinatorics. This coefficient represents the number of ways to choose \(k\) elements from a set of \(n\) elements, and it’s calculated using the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] where \(!\) denotes a factorial, meaning you multiply all whole numbers up to that number. For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Combinatorics not only helps in finding binomial coefficients efficiently but also extends to various applications, such as:

• Determining possible outcomes in probability.
• Optimizing configurations in optimization problems.
• Designing experiments that involve combinations without repetition.

Understanding combinatorics enriches our comprehension of the binomial expansion and provides a foundational tool for solving many mathematical and real-world problems.