Problem 47 - College Algebra with Corequisite Support

Question

Consider the expansion of \((x+b)^{40} .\) What is the exponent of \(b\) in the \(k\) the term?

Step-by-Step Solution

Answer

Exponent of \(b\) in the \(k\)th term is \(k\).

Step 1: Understand Binomial Expansion

The binomial theorem states that \((x+b)^n\) can be expanded using the formula \((x+b)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} b^k\). This implies we're dealing with terms of the form \(\binom{n}{k} x^{n-k} b^k\) where \(n = 40\) for this specific problem.

Step 2: Identify the Terms in the Expansion

From the expansion formula, each term in the expansion has the form \(\binom{n}{k} x^{n-k} b^k\). We are interested in the exponent of \(b\) in the \(k\)th term of this expansion.

Step 3: Write the General Form of a Term

Considering the formula \(\binom{40}{k} x^{40-k} b^k\), the \(k\)th term in the expansion of \((x+b)^{40}\) is \(\binom{40}{k} x^{40-k} b^k\). The exponent of \(b\) in the \(k\)th term is \(k\).

Step 4: Conclusion

Therefore, the exponent of \(b\) in the \(k\)th term of the expansion of \((x+b)^{40}\) is \(k\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful mathematical tool, allowing us to express powers of binomials in an expanded form. When you encounter an expression like \((x+b)^n\), the binomial theorem helps break it down into a sum of terms. By using the formula \((x+b)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} b^k\), you can calculate each term individually.

• The formula provides a way to expand binomials, which are expressions of two terms, raised to any power.
• Each term in this expansion includes binomial coefficients, powers of \(x\), and powers of \(b\).

By this method, a binomial raised to a high power is systematically expanded into simpler terms. The utility of binomial expansion is immense, allowing for simplified calculations in algebra, probability, and more.

Exponents in Binomial Expansion

In a binomial expansion, the exponents play a critical role. They indicate how many times a certain term is multiplied by itself. For the term \(b^k\) found in the expansion of \((x+b)^{40}\), the exponent \(k\) shows how many times \(b\) appears in the term.

• Each term in the binomial expansion of \((x+b)^n\) has a specific structure: \(x^{n-k}\) followed by \(b^k\).
• The exponents \(n-k\) and \(k\) vary with each change in \(k\), which represents the position of the term in the sequence.

Understanding how exponents vary is crucial when you solve problems involving binomial expansions, ensuring accurate representations of terms.

Combinatorial Coefficients

Combinatorial coefficients are an essential part of binomial expansions and are found using the binomial coefficient \(\binom{n}{k}\). These coefficients are numbers that describe how items can be selected or arranged.

• In the binomial expansion, \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from \(n\) without regard to order.
• This coefficient is calculated using factorials: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

These coefficients multiply the terms \(x^{n-k}b^k\) in the expansion, dictating the weight each term carries in the final expression. Combinatorial coefficients provide a foundation for understanding probability and statistics, making them incredibly valuable.