Problem 48 - College Algebra with Modeling and Visualization

Question

Find the specified term. The third term of \((a+b)^{7}\)

Step-by-Step Solution

Answer

The third term is \(21a^5b^2\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This means each term in the expansion is given by \(\binom{n}{k} a^{n-k} b^k\).

Step 2: Identify the Required Term

We need to find the third term of the expansion. The formula for the \((k+1)\)-th term is \(\binom{n}{k} a^{n-k} b^k\). For the third term, \(k = 2\) (since \(k\) starts from 0).

Step 3: Plug Values into the Formula

For \((a+b)^7\) and the third term, we have \(n = 7\) and \(k = 2\). Substitute into the term formula: \(\binom{7}{2} a^{7-2} b^2\).

Step 4: Calculate the Binomial Coefficient

Calculate \(\binom{7}{2}\): \(\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21\).

Step 5: Compute the Third Term

Substitute the binomial coefficient into the term expression: \(21 \times a^{5} \times b^2\). Therefore, the third term is \(21a^5b^2\).

Key Concepts

Binomial Coefficient

When dealing with polynomial expansions using the Binomial Theorem, the binomial coefficient plays a crucial role. A binomial coefficient is a numerical value that represents the coefficients of the expanded terms. You often see them represented in the form \( \binom{n}{k} \), pronounced "n choose k," where \( n \) is the total number of terms to choose from and \( k \) is the number of terms being chosen. These coefficients are found using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
- **\( ! \) stands for factorial**, which means multiplying a series of descending natural numbers. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This formula is used to calculate the number of combinations in the polynomial expansion.
The binomial coefficient helps determine the numerical multiplier of each term in a polynomial expansion. In our solution, we calculated the coefficient as \( 21 \) for the third term in the polynomial expansion \((a+b)^7\). Understanding how to derive this value is key for accurately constructing expanded polynomial expressions.

Polynomial Expansion

The process of polynomial expansion involves expressing a binomial expression raised to a power as a sum of individual terms. This is where the Binomial Theorem comes into play:
- It provides a formula for expanding the expression \((a+b)^n\) as a sum. Each term in this sum takes the form \( \binom{n}{k} a^{n-k} b^k \). The polynomial expansion is essentially listing out all possible products of the binomial's terms without performing the direct multiplication.
For example, when finding the third term of \((a+b)^7\), you don't manually multiply \((a+b)\) seven times. Instead, you rely on the formula to efficiently find each term. In our example, the third term is given by the expression \( \binom{7}{2} a^5 b^2 \), which expands to \( 21a^5b^2 \). Utilizing this theorem simplifies the process considerably.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. In algebra, we use expressions to represent quantities and their relationships. When we talk about a binomial, specifically it refers to an algebraic expression consisting of two terms, like \( a + b \). The power of algebraic expressions comes into play when they are raised to higher powers, as seen in binomial expressions.
The algebraic expression \((a+b)^n\) involves two key operations: addition and exponentiation. Each term in the expanded form derives from these operations applied systematically across the binomial's parts.
- **Variables** like \( a \) and \( b \) can hold any number, which allows algebraic expressions to be highly flexible and adaptable. - **Operations** such as raising a binomial to a power demonstrate the application of rules like distributing power over addition.
In the example \((a+b)^7\), the algebraic nature of the expression allows us to compute specific terms, such as \( 21a^5b^2 \), by using the established rules of algebra. Understanding these expressions and their manipulation is fundamental in algebra, providing the tools to solve complex mathematical problems.