Problem 23 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(c+2)^{5}$$

Step-by-Step Solution

Answer

The simplified form of the expression \((c+2)^5\) using the Binomial Theorem is \(c^5 + 10c^4 + 40c^3 + 80c^2 + 80c + 32\).

Step 1: Apply Binomial Theorem

The binomial theorem states that for any real number \(a\) and \(b\), \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^{k}\). Apply this theorem to the given problem \((c+2)^{5}\).

Step 2: Expand binomial expression

Expansion using the Binomial Theorem gives us the following: \((c+2)^5 = \binom{5}{0}c^{5}2^{0} + \binom{5}{1}c^{4}2^{1} + \binom{5}{2}c^{3}2^{2} + \binom{5}{3}c^{2}2^{3} + \binom{5}{4}c^{1}2^{4} + \binom{5}{5}c^{0}2^{5}\).

Step 3: Simplify the expression

To simplify, calculate the binomial coefficients and the powers of 2. Then multiply each term by their respective multiples to get the final result: \(c^5 + 10c^4 + 40c^3 + 80c^2 + 80c + 32\)

Key Concepts

Binomial Coefficients

Binomial coefficients are vital components in the Binomial Theorem and have a significant presence in combinatorics. They are the numbers that appear as the coefficients in the binomial expansion. To understand them, we can look at the notation \( \binom{n}{k} \), known as 'n choose k,' which represents the number of ways to pick k items out of n without considering the order. For a given exponent \(n\) in a binomial expansion, the coefficients will start with \( \binom{n}{0} \) for the first term and end with \( \binom{n}{n} \) for the last term, with \(k\) increasing by 1 in each subsequent term.

For example, in the polynomial \( (c+2)^5 \), the binomial coefficients can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). It results in 1, 5, 10, 10, 5, and 1 as the coefficients for the expansion terms, representing the number of combinations to pick k elements from a five-element set.

Polynomial Expansion

The process of expanding a polynomial expression involves converting it from its compact \( (a + b)^n \) form into a series of terms without parentheses, using the properties of exponents and combinations. The Binomial Theorem aids this expansion by providing a systematic way to determine all the terms and their coefficients.

When expanding \( (c+2)^5 \), we apply the theorem as illustrated in the exercise. Each term after the expansion is a product of a binomial coefficient, a power of \(c\), and a power of 2. The expanded form showcases the polynomial as a sum of monomials.

Understanding the Expanded Form

The final expanded form lays out each individual term created from the application of the Binomial Theorem. It reveals the structure of the polynomial and provides a clear picture of how each binomial coefficient and variable exponent combine to form the terms of the expanded binomial expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. In the context of binomial expansions, these expressions become more complex as we deal with polynomials.

The Binomial Theorem transforms a simple algebraic expression, such as \( (c+2)^5 \), into a far more intricate one by detailing the relationship between its terms through expansion. Simplifying the algebraic expression involves calculating the powers and the products of the binomial coefficients with the corresponding variable terms. The expression \(c^5 + 10c^4 + 40c^3 + 80c^2 + 80c + 32\) shows the result of this process after simplification - each term clearly displays the interplay between the constant, the variable, and their respective exponents.

Critical Relevance of Simplification

By simplifying the expression, we not only make the algebraic representation cleaner but it also becomes more usable for further mathematical operations such as differentiation, integration, factorization, or solving equations.