Problem 31 - College Algebra

Question

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x+2)^{8}$$

Step-by-Step Solution

Answer

The first three terms in the binomial expansion of \((x+2)^8\) are \(x^8\), \(16x^7\), and \(112x^6\).

Step 1: Determine the first term

The first term is obtained by applying the binomial coefficient to x^8 and 2^0. The binomial coefficient for k=0 is (8 choose 0), which is 1. Thus, the first term is \(1*x^8*2^0\) or \(x^8\).

Step 2: Determine the second term

The second term is obtained by applying the binomial coefficient to x^7 and 2^1. The binomial coefficient for k=1 is (8 choose 1), which is 8. Therefore, the second term is \(8*x^7*2^1\) or \(16x^7\).

Step 3: Determine the third term

The third term is calculated by applying the binomial coefficient to x^6 and 2^2. The binomial coefficient for k=2 is (8 choose 2) equals 28. Thus, the third term is \(28*x^6*2^2\) or \(112x^6\).

Key Concepts

Binomial Theorem

The binomial theorem provides a systematic method to expand expressions that are raised to a power, like \( (x + 2)^8 \). It expresses a binomial raised to a power as the sum of terms involving coefficients, powers, and products.
A binomial expression is of the form \( (a + b)^n \). To expand it, you need to understand how the terms are structured:

• Each term includes a binomial coefficient.
• The exponents of the first term decrease, while the exponents of the second term increase, all adding up to \( n \).
• Binomial coefficients can be found using combinations.

Understanding this theorem simplifies the process, allowing you to efficiently compute each term without repetitive multiplication.
Using this method, you can quickly determine terms by plugging in values for \( k \) in the combination formula \( \binom{n}{k} \) to get the powers and coefficients.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of sets. In the context of the binomial theorem, it helps us determine the binomial coefficients. These coefficients are essential for finding each term in the expansion.
The binomial coefficient \( \binom{n}{k} \) is calculated as follows:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
• It represents the number of ways to choose \( k \) elements from a set of \( n \) elements.
• "n!" means the product of all positive integers up to \( n \), known as "n factorial".

These coefficients play a crucial role in each term of a binomial expansion. Knowing how to compute them lets you explore higher powers of binomials easily.

Algebraic Expressions

Algebraic expressions are composed of variables, constants, and operators. In algebra, when you expand expressions like \( (x + 2)^8 \), you translate a compact expression into a polynomial form.
The steps typically involve:

• Identifying base terms and their powers.
• Applying operations like multiplication and addition.
• Simplifying each term by performing operations in the correct order.

However, using the binomial theorem, algebraic expressions are expanded systematically, utilizing the concept of expansions and combinations, which makes the task much more manageable.
Simplification is key, as shown in the exercise, where coefficients and powers are combined to form clear, simplified terms such as \( x^8 \), \( 16x^7 \), and \( 112x^6 \). This enhances your ability to work with complex algebraic expressions efficiently.