Problem 36 - Precalculus: Mathematics for Calculus

Question

Find the indicated terms in the expansion of the given binomial. The 28 th term in the expansion of \((A-B)^{30}\).

Step-by-Step Solution

Answer

The 28th term is \(-4060 A^3 B^{27}\).

Step 1: Review of Binomial Expansion Formula

The binomial theorem is given by: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is a binomial coefficient. For expansion \((A-B)^{30}\), we set \(a = A\), \(b = -B\), and \(n = 30\).

Step 2: Understanding Term Requirement

The general term in the expansion of \((A-B)^{30}\) is \(T_{k+1} = \binom{30}{k} A^{30-k} (-B)^k\). We are tasked with finding the 28th term; therefore, we need \(T_{28}\).

Step 3: Determine the Value of k

Since we want the 28th term, we set \(k+1 = 28\), which implies \(k = 27\).

Step 4: Calculate the 28th Term

Substitute \(k = 27\) into the general term formula: \[T_{28} = \binom{30}{27} A^{30-27} (-B)^{27}\].

Step 5: Compute Binomial Coefficient

Calculate \(\binom{30}{27}\) using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\): \[\binom{30}{27} = \frac{30!}{27! \, 3!} = 4060\].

Step 6: Final Expression of the Term

Now, substitute the binomial coefficient back into the term equation: \[T_{28} = 4060 \cdot A^3 \cdot (-B)^{27} = 4060 \cdot A^3 \cdot (-1)^{27} \cdot B^{27} = -4060 A^3 B^{27}\].

Key Concepts

Understanding the Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions of the form \((a+b)^n\). It allows us to express the expanded form as a sum of terms involving both \(a\) and \(b\). This theorem tells us that:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here’s a quick breakdown of what this formula means:

• The sum \(\sum\) runs from \(k=0\) to \(k=n\).
• Each term in the sum involves a binomial coefficient, \(\binom{n}{k}\), which tells us how many ways we can choose \(k\) elements from a set of \(n\) elements.
• The terms \(a^{n-k}\) and \(b^k\) represent the contributions of \(a\) and \(b\) in each term, with their exponents adding up to \(n\).

The binomial theorem is particularly helpful when dealing with polynomials raised to large powers, like in our original problem of \((A-B)^{30}\). Knowing how to apply this theorem can simplify the expansion process greatly.

Calculating the Binomial Coefficient

The binomial coefficient \(\binom{n}{k}\) is a crucial part of the binomial theorem, representing the number of ways to choose \(k\) items from a set of \(n\) without regard to order. It is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here’s a quick guide to understand how to compute it:

• \(n!\) (n factorial) is the product of all positive integers up to \(n\).
• \(k!\) is the product of all positive integers up to \(k\).
• \((n-k)!\) is the factorial of the difference \(n-k\).

In the original exercise, the coefficient \(\binom{30}{27}\) was calculated as follows:\[\binom{30}{27} = \frac{30!}{27! \times 3!} = 4060\]This calculation is essential to determine the multiplicative factor of the term in question.

Identifying Expansion Terms

When expanding a binomial like \((A-B)^{30}\), each term represents a specific combination of \(A\) and \(-B\). These terms are formulated using the general term:\[T_{k+1} = \binom{n}{k} a^{n-k} b^k\]For our exercise, this translates to:\[T_{k+1} = \binom{30}{k} A^{30-k} (-B)^k\]Identifying expansion terms involves determining the correct value of \(k\) for the desired term. For example, if you want to find the 28th term in the expansion, set \(k+1 = 28\), which gives \(k = 27\). The term then becomes:\[T_{28} = \binom{30}{27} A^3 (-B)^{27}\]To complete the term's computation, substitute known values like the binomial coefficient calculated earlier.

Using Pascal's Triangle

Pascal's Triangle is a simple, yet effective tool to find binomial coefficients without complex calculations. Each row in the triangle represents the coefficients when a binomial is expanded. Row zero reflects \((a+b)^0\), row one is \((a+b)^1\), and so on. To retrieve the binomial coefficient \(\binom{n}{k}\), simply look at the \(k^{th}\) position in the \(n^{th}\) row.
For instance, in calculating \(\binom{30}{27}\), find the 27th entry in the 30th row (taking into account rows and entries start at zero).
Pascal's Triangle helps students verify calculated coefficients quickly and visually. It's a practical way to cross-check your calculations, especially with large values of \(n\). This triangle not only simplifies finding binomial coefficients but also enhances understanding of the underlying algebraic concepts.