Problem 48 - Algebra 2

Question

Find the specified term of each binomial expansion. Third term of \((x+3)^{12}\)

Step-by-Step Solution

Answer

The third term of the expansion of \((x+3)^{12}\) is \(118098x^{10}\).

Step 1: Understand The Binomial Coefficient

The binomial coefficient for the term \(k+1\) in the expansion of \((a+b)^n\) can be given by the combination \(_nC_k\). It's defined as \(_nC_k = \frac{n!}{k!(n-k)!}\) where \(n!\) is the factorial of \(n\), meaning \(n \cdot (n-1) \cdot (n-2) \cdot \ldots \cdot 2 \cdot 1\). For this exercise the binomial is \((x+3)^{12}\) and we are looking for the third term of the expansion, so \(_nC_k = _{12}C_{2}\).

Step 2: Compute the Binomial Coefficient

Now, calculate the combination \(_{12}C_{2}\). Using the formula for combinations, we know that \(_{12}C_{2} = \frac{12!}{2!(12-2)!}\). This simplifies to \(_{12}C_{2} = 66\).

Step 3: Compute the Coefficient of the Third Term

The specific term in the binomial expansion is given by \(t_{k+1} = _nC_k * a^{n-k} * b^k\), where \(a\) and \(b\) are the terms of the binomial. Here, \(k=2\), \(n=12\), \(a=x\) and \(b=3\). Therefore, \(t_{3} = _{12}C_{2} * x^{12-2} * 3^2 = 118098\).

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental concept in algebra, allowing us to expand expressions raised to a power, specifically of the form \((a+b)^n\). This expansion results in a series of terms where each term includes a binomial coefficient, a power of \(a\), and a power of \(b\). The theorem can be represented as: \[(a+b)^n = \sum_{k=0}^{n} \_nC_k \cdot a^{n-k} \cdot b^k.\]

• The symbol \(\_nC_k\) represents a binomial coefficient and determines how many ways you can select \(k\) items from a total of \(n\).
• The theorem is especially useful as it can simplify the calculation of terms in polynomial expansions by not requiring manual multiplication across all combinations.
• Each term in the expansion of \((a+b)^n\) has a structure that follows the pattern determined by the theorem, making it predictable.

Combinatorics

Combinatorics is the area of mathematics dealing with counting, arrangement, and combination. It's a crucial tool when working with binomials and plays a key role in understanding the coefficients in binomial expansions.

• The concept of \(\_nC_k\), often read as "n choose k," originates from combinatorics.
• These choices are calculated using combinations, and this concept helps determine the specific term in a binomial expansion.
• Combinatorics bridges the gap between straightforward counting and more complex algebraic manipulations in math problems like our binomial expansion.

In our exercise, \(\_nC_k\) was used to determine how we chose which term to expand first between the factors of \((x+3)^{12}\).

Factorials

Factorials are an important component in calculating combinations and permutations in mathematics. The notation \(n!\) (read as "n factorial") is used to represent the product of all positive integers up to \(n\). It's defined mathematically as:

• \(n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1\)
• The factorial is integral to the formula for combinations: \(_nC_k = \frac{n!}{k!(n-k)!}\).
• In our example, calculating \(\_{12}C_{2}\) involves the factorials of \(12\), \(2\), and \(10\). This calculation simplifies the process of finding specific terms in polynomial expansions.

Understanding factorials is essential when you work with binomial expansions as it allows you to compute the necessary coefficients efficiently.