Problem 36 - Algebra and Trigonometry

Question

Find the 28 th term in the expansion of \((A-B)^{30}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Binomial Expansion

The binomial theorem allows us to expand expressions of the form \((A-B)^n\) into a sum of terms involving binomial coefficients. It states that \((A-B)^n = \sum_{k=0}^{n} \binom{n}{k} A^{n-k} (-B)^k\). Each term in this expansion is of the form \(\binom{n}{k} A^{n-k} (-B)^k\).

Step 2: Determine the General Term

The general term (also known as the \(k+1\)-th term) in the expansion of \((A-B)^n\) is given by \(T_{k+1} = \binom{n}{k} A^{n-k} (-B)^k\). For \(n = 30\), the term becomes \(T_{k+1} = \binom{30}{k} A^{30-k} (-B)^k\).

Step 3: Identify the Term Number

We are asked to find the 28th term in the expansion. In terms of the general term formula, this corresponds to \(k = 27\) because \(T_{k+1} = T_{28}\) means \(k = 28 - 1 = 27\).

Step 4: Substitute the Values

Substitute \(n = 30\) and \(k = 27\) into the general term expression: \(T_{28} = \binom{30}{27} A^{30-27} (-B)^{27}\).

Step 5: Calculate the Binomial Coefficient

Calculate the binomial coefficient \(\binom{30}{27}\). The property of binomial coefficients states that \(\binom{n}{k} = \binom{n}{n-k}\), so \(\binom{30}{27} = \binom{30}{3}\). Calculating gives \(\binom{30}{3} = \frac{30 \times 29 \times 28}{3 \times 2 \times 1} = 4060\).

Step 6: Apply the Exponents

Substitute in the calculated coefficient: \(T_{28} = 4060 \cdot A^{3} \cdot (-B)^{27}\). Simplify to get \(T_{28} = 4060 \cdot A^{3} \cdot (-1)^{27} \cdot B^{27}\), which simplifies to \(T_{28} = -4060A^3B^{27}\) since \((-1)^{27} = -1\).

Key Concepts

Binomial Theorem

The Binomial Theorem is an essential concept in algebra that facilitates the expansion of expressions raised to a power. Specifically, it applies to expressions of the form \((A+B)^n\) or \((A-B)^n\). This theorem states that:

• A binomial expression like \((A-B)^n\) can be expanded into a sum of terms.
• Each term involves a coefficient known as the binomial coefficient.
• The formula for this expansion is given by \((A-B)^n = \sum_{k=0}^{n} \binom{n}{k} A^{n-k} (-B)^k\).

The significance of this theorem lies in its ability to simplify the computation of high powers. Rather than multiplying directly, which is tedious and error-prone, the binomial theorem provides a structured pathway for polynomial expansion.
It also reveals interesting mathematical patterns, such as Pascal's Triangle, which graphically represents the coefficients. Understanding how to apply the theorem is foundational for tackling problems in algebra and calculus.

Binomial Coefficient

The binomial coefficient is a crucial component of the binomial theorem. It helps determine the weight or impact of each term in an expanded binomial expression. The binomial coefficient is denoted by \(\binom{n}{k}\), which reads as "n choose k". This coefficient is calculated using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

where \(n!\) (n factorial) is the product of all positive integers up to \(n\).
For example, in the expression \((A-B)^{30}\), if we want to find the 28th term using \(k=27\), the coefficient \(\binom{30}{27}\) simplifies to \(\binom{30}{3}\) due to the symmetry property \(\binom{n}{k} = \binom{n}{n-k}\). This can be calculated as \(4060\), showing the impactful role these coefficients play in determining the magnitude and sign of each term. Understanding how to compute these coefficients is vital for accurate polynomial expansion.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial, like \((A-B)^{30}\), in an extended form where each term is a product of powers of the individual variables. The purpose of the expansion is to transform complex expressions into a more workable form. With the Binomial Theorem, the process of expanding polynomials becomes systematic and straightforward.

• Each term in the expansion is constructed by determining the binomial coefficient and applying the exponents to \(A\) and \(-B\).
• The expansion clarifies the impact of each variable and its power on the overall expression.
• In the example of finding the 28th term, the expansion provides the structured approach: \(T_{28} = -4060A^3B^{27}\).

Polynomial expansion isn't just limited to computations; it offers insights into grouping terms and solving algebraic identities. By breaking down the terms, students can better understand how the elements interact and affect one another.