Problem 30 - Algebra and Trigonometry

Question

Find the fifth term in the expansion of \((a b-1)^{20}\)

Step-by-Step Solution

Answer

The fifth term is \(4845 (a b)^{16}\).

Step 1: Identify the General Term of the Binomial Expansion

The general term in the expansion of \((a b - 1)^{20}\) using the binomial theorem is given by the formula: \[T_{k+1} = \binom{n}{k} (a b)^{n-k} (-1)^k\] where \(n = 20\) and \(k\) is the term index starting from 0.

Step 2: Determine the Position of the Fifth Term

To find the fifth term, we use \(k = 4\). This is because the term position is given by \(k + 1\), so for the fifth term, \(k = 4\).

Step 3: Plug in the Values to the Binomial Term

Substitute \(n = 20\) and \(k = 4\) into the general term formula:\[T_{5} = \binom{20}{4} (a b)^{20-4} (-1)^4\] Simplifying, we get:\[T_{5} = \binom{20}{4} (a b)^{16}\]

Step 4: Calculate the Binomial Coefficient

Calculate \(\binom{20}{4}\):\[\binom{20}{4} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845\]

Step 5: Construct the Fifth Term

Using the value from the binomial coefficient, substitute back:\[T_{5} = 4845 \times (a b)^{16}\]Therefore, the fifth term in the expansion is \(4845 (a b)^{16}\).

Key Concepts

Binomial Expansion

The binomial expansion is a way of expressing expressions raised to a power, such as \((a + b)^n\), in an expanded form. It allows you to write the original expression as a sum of terms involving coefficients, the variables in the binomial, and their exponents.

• Each term in the expansion is formed by taking the product of a binomial coefficient and the variables raised to their respective powers.
• The exponents of the variables depend on the position of the term in the expansion.
• Specifically, if we expand \((a + b)^n\), each term is of the form \(\binom{n}{k}a^{n-k}b^k\) where \(k\) ranges from 0 to \(n\).

In problems like expanding \((ab-1)^{20}\), the binomial expansion gives us a systematic way to find any term, such as the fifth term, by applying the binomial theorem formula. This concept is particularly useful in probability, statistics, and algebra, where such expansions provide detailed analysis of polynomial expressions.

Binomial Coefficient

The binomial coefficient is a key part of the binomial theorem and is denoted as \(\binom{n}{k}\). It is a way of selecting \(k\) items from \(n\) items without regard to the order of selection.

• The binomial coefficient is calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• Factorials, denoted by \(!\), are used in the formula, representing the product of all positive integers up to \(n\).
• It measures how many ways you can combine different parts of the expression in each term of the binomial expansion.

In the exercise where we find the fifth term of \((ab-1)^{20}\), calculating the binomial coefficient \(\binom{20}{4}\) was essential. This gave us the number "4845," indicating how many different ways we can combine parts of \(ab\) raised to the remaining power.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, arrangements, and counting. It provides the tools to handle selections from sets, permutations, and how items can be grouped or ordered under specific constraints.

• It is essential in understanding the binomial theorem since binomial coefficients are fundamentally combinations.
• Combinatorial methods help in determining how terms in binomial expansions are structured in terms of their order and position.
• The combination formula \(\binom{n}{k}\) arises directly from combinatorics, as it counts the number of ways \(k\) items can be chosen from \(n\).

Through these principles, we calculate specific terms in expanded polynomial expressions, improving our ability to analyze and predict outcomes in various mathematical scenarios.

Algebraic Expressions

Algebraic expressions involve a combination of variables, constants, and operations (addition, subtraction, multiplication, and division). These expressions form the foundation of understanding how terms are expanded in algebraic calculations like binomial expansions.

• An algebraic expression is a sum of terms, each consisting of a coefficient and a variable raised to a power.
• In the binomial expansion, each term of the expression contributes to the final polynomial result.
• Managing algebraic expressions efficiently involves sifting through variables and coefficients to identify patterns, simplifying them, or solving equations.

When working on an expansion such as \((ab-1)^{20}\), the algebraic expression guides how each term in the expansion is computed, involving both the variable \(ab\) and the constant \(-1\) raised to the power related to the term index. Mastery of these expressions is crucial to solving and simplifying complex algebraic problems effectively.