Problem 31 - College Algebra

Question

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 \(x^8, 16x^7 ,112x^6.\)

Step 1: Identification of the Binomial Expressions

Here, the binomial expressions are \(x\) and \(2)\. The power \(n = 8\). So the expansion will have \(8+1 = 9\) terms.

Step 2: Finding the Binomial Coefficients

The binomial coefficients for the first three terms are:- The 1st term: \( ^{8}C_{0} = 1 \)- The 2nd term: \( ^{8}C_{1} = 8 \)- The 3rd term: \( ^{8}C_{2} = 28 \)

Step 3: Formulation of the First Three Terms of the Binomial Expansion

The first three terms of the binomial expansion are determined by applying the binomial formula:- The 1st term: \(1 * x^8 * 2^0 = x^8\)- The 2nd term: \(8 * x^7 * 2^1 = 16x^7\)- The 3rd term: \(28 * x^6 * 2^2 = 112x^6\)Therefore the first three terms of the binomial expansion, in simplified form, are \(x^8, 16x^7\) and \(112x^6\) respectively.

Key Concepts

Binomial Theorem

The Binomial Theorem is a cornerstone of algebra that allows us to expand expressions raised to a power in the form of \( (a+b)^n \). It provides a systematic method for expressing the expanded form as a sum of terms involving binomial coefficients. These binomial coefficients form the numbers known as Pascal's triangle. Each term in the binomial expansion is a combination of powers of both \(a\) and \(b\), with the exponents of \(a\) decreasing and those of \(b\) increasing in each successive term.

For example, in the expansion of \( (x+2)^8 \), which involves eight applications of the binomial, the theorem lets us find the first three terms systematically without fully expanding all terms. Using the binomial theorem, each term in the expansion is determined sequentially, using a formula that involves binomial coefficients—which, in turn, can be found using factorial notation or Pascal's triangle.

Binomial Coefficients

Binomial coefficients are the numerical factors that multiply the terms in a binomial expansion. These coefficients are commonly denoted as \( ^{n}C_{k} \) or \( C(n, k) \), and they articulate the number of ways to choose \(k\) elements out of a total of \(n\), regardless of the order. Essentially, they are a key part of the binomial theorem's formula: \[ (a+b)^n = \sum_{k=0}^{n} (^{n}C_{k})a^{n-k}b^{k} \]

The coefficients for any binomial expression can be calculated using the formula \( ^{n}C_{k} = \frac{n!}{k!(n-k)!} \), where \(n!\) denotes the factorial of \(n\) and represents the product of all positive integers up to \(n\). For instance, the binomial coefficients for the first three terms of the expansion \( (x+2)^8 \) are \( ^{8}C_{0}, ^{8}C_{1}, \) and \( ^{8}C_{2} \). These are found to be 1, 8, and 28, respectively, which align with our example. These coefficients are a crucial part of each term in the expanded binomial expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like \(x\) or \(y\)), and operators (such as addition and multiplication). The binomial expansion itself yields an extended algebraic expression of terms that are algebraically significant. These expressions can be simplified, factored, or manipulated using algebraic theorems and properties to solve equations or understand mathematical relationships.

In the context of binomial expansion, we turn the expression \( (x+2)^8 \) into a sum of simpler algebraic expressions that represent the terms of the expanded polynomial. Each term reflects the products of coefficients with variable \(x\) and the constant raised to respective powers. Evaluating these terms and arranging them in descending powers of \(x\), as we would with any polynomial expression, makes them more manageable and prepares us for further algebraic operations, whether they're for simple evaluation, graphing, or solving equations.