Problem 23 - Algebra 2

Question

Find the indicated term of each expansion. fourth term of \((x+2)^{7}\)

Step-by-Step Solution

Answer

The fourth term of \((x+2)^7\) is \(280x^4\).

Step 1: Understand the Binomial Theorem

The binomial theorem tells us how to expand expressions of the form \((a + b)^n\) as a series of terms. Each term in the expansion of \((x+2)^7\) can be determined using the formula \(\binom{n}{k} a^{n-k} b^k\) where \(a = x\), \(b = 2\), \(n = 7\), and \(k\) is the term number minus one.

Step 2: Identify the term number

We are tasked with finding the fourth term in the expansion of \((x+2)^7\). For the fourth term, we set \(k = 4 - 1 = 3\).

Step 3: Calculate the binomial coefficient

The binomial coefficient is given by \(\binom{n}{k}\), which for our problem is \(\binom{7}{3}\). This is calculated as \(\frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\).

Step 4: Substitute into the binomial formula

Now substitute \(n = 7\), \(k = 3\), \(a = x\), and \(b = 2\) into the formula: \(\binom{7}{3} x^{7-3} \cdot 2^3\). This becomes \(35 \cdot x^4 \cdot 8\).

Step 5: Simplify the expression

Calculate \(35 \cdot 8 = 280\), so the fourth term is \(280x^4\).

Key Concepts

Binomial Coefficients

Binomial coefficients are fundamental components of the Binomial Theorem. They are represented as \( \binom{n}{k} \), which is called the "n choose k" notation. These coefficients help us determine the number of ways to choose \( k \) items from \( n \) total items. In the context of polynomial expansion, binomial coefficients specify the number of ways each term's components can be arranged.

To compute a binomial coefficient, use the formula:

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

The exclamation point denotes a factorial, meaning you multiply all whole numbers from that number down to one. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Understanding binomial coefficients aids in the calculation of specific terms during polynomial expansions, allowing for an efficient and organized approach to solving such problems.

Polynomial Expansion

Polynomial expansion is a method to rewrite a binomial expression in a more manageable form, like transforming \((a + b)^n\) into a sum of terms. Each term involves powers of the original components, multiplied by binomial coefficients. This process is governed by the Binomial Theorem, which provides a systematic approach to expand binomial expressions.

Key characteristics of polynomial expansion include:

• The number of terms in the expanded form matches \( n+1 \), where \( n \) is the exponent in the binomial expression \((a + b)^n\).
• Each term follows the general form: \( \binom{n}{k} a^{n-k} b^k \), where \( k \) varies from 0 to \( n \).

In our example, \((x+2)^7\), polynomial expansion helps break down the expression into pieces that are easier to evaluate or compute. This breakdown involves combining powers of \( x \) and 2, scaled by binomial coefficients, thus offering a structured way to handle more complex algebraic expressions.

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arrangement, and combination of elements. This discipline underlies various aspects of binomial expansions, especially when it involves calculating the binomial coefficients, which quantify combinations.

In the binomial theorem, combinatorics helps compute the various ways terms can be arranged in any given order. Essentially, each term in a binomial expansion is a product of a binomial coefficient (a combination of elements) with powers of the individual elements.

Through understanding combinatorics, students can appreciate the logical structures behind combinations and the rationale of using binomial coefficients in polynomial expansions. Important aspects include:

• Understanding how factorials relate to permutations and combinations.
• Realizing that combinatorial principles allow for efficient calculation rather than listing all possibilities.

Recognizing the role of combinatorics in solving binomial problems reinforces the student's ability to analyze and tackle a wide variety of mathematical scenarios.