Problem 53 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. (w+1)^{15} ; \text { tenth term }

Step-by-Step Solution

Answer

The tenth term of the binomial expansion of \((w+1)^{15}\) is \(5005w^6\).

Step 1: Identify the binomial theorem formula for coefficients

The formula for finding the binomial coefficients in a binomial expansion is: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

Step 2: Use the formula to find the tenth term's coefficient

Plug in the given values, \(n = 15\) and \(k = 9\), into the formula: \[\binom{15}{9} = \frac{15!}{9!(15-9)!} = \frac{15!}{9!6!}\] Now, we calculate the factorial and simplify: \[\binom{15}{9} = \frac{15\times 14 \times 13 \times 12\times 11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{(9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1)(6\times 5\times 4\times 3\times 2\times 1)}\] Cancel out common terms from the numerator and denominator: \[\binom{15}{9} = \frac{15\times 14 \times 13 \times 12\times 11\times 10}{5\times 4\times 3\times 2\times 1}\] Now, continue to simplify the expression: \[\binom{15}{9} = \frac{15\times 14 \times 13 \times 12\times 11\times 10}{120} = 5005\]

Step 3: Write the tenth term using the coefficient

Now we have the coefficient for the tenth term: 5005. Recall that the binomial expansion follows the form \((a+b)^n\), and in our case, \(a=w\), \(b=1\), and \(n=15\). Using the general formula for the term in a binomial expansion, which is \(\binom{n}{k}a^{n-k}b^k\), and applying it to our case, we have: \[\text{Tenth term} = 5005 \cdot w^{15-9} \cdot 1^9\] This simplifies to: \[\text{Tenth term} = 5005w^6\] The tenth term of the binomial expansion of \((w+1)^{15}\) is \(5005w^6\).

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that provides a way to expand expressions raised to any power. When we talk about a binomial expression, we're working with something like \((a + b)^n\). The theorem gives us a systematic means to expand this expression into a sum of terms of the form \(\binom{n}{k} a^{n-k} b^k\).
This expansion results in coefficients which are drawn from binomial coefficients \(\binom{n}{k}\), determined through a combinatorial approach. The theorem simplifies solving problems by allowing us to find specific terms without fully expanding the entire expression.

Factorial Calculation

The factorial is an essential concept in mathematics, especially when dealing with permutations, combinations, and the binomial theorem. It is denoted by an exclamation mark (\(!\)) and is used to signify the product of an integer and all integers below it. For example, \(n!\) means \(n \times (n-1) \times (n-2) \times ... \times 1\).
Factorials grow incredibly fast, which can cause computational challenges. However, their use simplifies the representation of large numbers and equations, especially in binomial coefficients. To solve equations involving binomials, you often need to simplify factorials—canceling common terms between the numerator and denominator makes calculations more manageable.

Mathematical Coefficients

Mathematical coefficients in the context of binomials refer to the binomial coefficients \(\binom{n}{k}\). These coefficients are integral to the arrangement of terms in a binomial expansion. Each coefficient is calculated using the formula:

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

These coefficients tell us how many ways we can choose \(k\) successes from \(n\) trials, which is analogous to selecting powers in a binomial expansion. The symmetric nature of these coefficients, as seen in Pascal's Triangle, means \(\binom{n}{k} = \binom{n}{n-k}\), displaying a connection between choosing \(k\) elements and the complementary \(n-k\) elements.