Problem 34 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Binomial Theorem

To find the fifth term of the binomial expansion of \((ab-1)^{20}\), we use the binomial theorem. The theorem states that for any positive integer \(n\), \((x + y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\). This implies that we need to find the general term first.

Step 2: Write the General Term

The general term \(T_{k+1}\) in the expansion \((ab-1)^{20}\) is given by \(T_{k+1} = \binom{20}{k} (ab)^{20-k} (-1)^{k}\). The negative sign comes from replacing \(y\) with \(-1\).

Step 3: Determine the Specific Term Index

We need the fifth term, so we set \(k+1 = 5\), thus \(k = 4\).

Step 4: Calculate Binomial Coefficient

Calculate the binomial coefficient: \(\binom{20}{4} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845\).

Step 5: Substitute Values into the General Term

Substitute \(k = 4\) into the general term: \(T_5 = \binom{20}{4} (ab)^{16} (-1)^{4} = 4845 (ab)^{16} (1)\).

Step 6: Simplify the Result

Finally, simplify the expression: \(T_5 = 4845 (ab)^{16}\).

Key Concepts

Binomial Expansion

The binomial expansion is an essential concept in algebra that allows us to expand expressions raised to a power, specifically expressions in the form \((x + y)^{n}\). This powerful tool provides a systematic method for expanding polynomials. The expansion is expressed as the sum of terms involving coefficients, powers of \(x\), and powers of \(y\).

One of the main applications of the binomial expansion is finding specific terms in large power expressions, like \((a b - 1)^{20}\). In this case, instead of manually multiplying the expression 20 times, we use the binomial theorem to find any desired term directly. This saves an immense amount of time and effort.

The general formula involves the binomial coefficients and can be expressed as:

• \[ (x + y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k} \]

If you have an expression like \((ab - 1)^{20}\), the expansion lets you systematically dissect the formula into separate terms by assigning each component in expressions a place according to the binomial theorem. This approach unpacks potentially overwhelming expressions with mathematically effortless precision.

Binomial Coefficient

In polynomial expressions, the binomial coefficient is a crucial element of the binomial expansion accompanied by combinations. It quantifies the number of ways to choose \(k\) elements from \(n\) elements without regard to the order, represented by \(\binom{n}{k}\). This value is obtained using the formula:

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

In our example, to find the fifth term in the expansion of \((ab - 1)^{20}\), we need the coefficient \(\binom{20}{4}\), because we applied the binomial theorem with \(k+1 = 5\). By substituting into the formula, we find:

• \[ \binom{20}{4} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845\]

This coefficient tells us how many ways we can choose 4 elements from 20, illustrating its foundational role in breaking down expressions such as the one we have into manageable pieces.

General Term

The general term in a binomial expansion provides an efficient way to determine any specific term without fully expanding the polynomial expression. Defined by the formula:

• \[ T_{k+1} = \binom{n}{k} x^{n-k} y^{k} \]

For the expression \((ab - 1)^{20}\), the general term was given by:

• \[ T_{k+1} = \binom{20}{k} (ab)^{20-k} (-1)^{k} \]

This reflects the power to which each part of the binomial is raised and how they contribute to their respective terms.

To find the fifth term, set \(k+1 = 5\), solving for \(k\), which gives \(k = 4\). Plugging this value into the general term formula gives us:

• \[ T_5 = \binom{20}{4} (ab)^{16} (-1)^{4} = 4845 (ab)^{16} (1) \]

Understanding the general term is invaluable for determining specific components of an expansion without analyzing each part manually.