Problem 24 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(c+3)^{5}$$

Step-by-Step Solution

Answer

The expanded form of \((c+3)^{5}\) using the Binomial Theorem is \(c^{5} + 45c^{4} + 90c^{3} + 270c^{2} + 405c + 243\)

Step 1: Identify the Variables

Identify the variables in the binomial expression. In this case, \(a=c\), \(b=3\), and \(n=5\).

Step 2: Apply the Binomial Theorem

Apply the Binomial Theorem, which states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). So, \((c+3)^5 = \sum_{k=0}^{5} \binom{5}{k} c^{5-k} 3^k\).

Step 3: Expand Terms

Expand the terms using the Binomial Theorem. \((c+3)^5 = \binom{5}{0} c^{5} 3^0 + \binom{5}{1} c^{4} 3^1 + \binom{5}{2} c^{3} Bonds3^2 + \binom{5}{3} c^{2} 3^3 + \binom{5}{4} c^{1} 3^4 + \binom{5}{5} c^{0} 3^5\)

Step 4: Simplify Terms

Simplify each term. The final simplified expression is \(c^{5} + 15c^{4}*3 + 10c^{3}*9 + 10c^{2}*27 + 5c*81 + 243\)

Step 5: Final Simplified Form.

Combine all terms to get the final simplified form. \(c^{5} + 45c^{4} + 90c^{3} + 270c^{2} + 405c + 243\)

Key Concepts

Binomial Expansion

Binomial expansion is a critical mathematical technique used to expand expressions that are raised to a power. Specifically, it deals with expressions of the form \((a + b)^n\). The expansion results in a sum of terms, each involving the binomial coefficients. These coefficients correspond to the numbers found in Pascal's Triangle. To expand, we use the Binomial Theorem: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). For instance, when we apply this theorem to \((c+3)^5\), we break down the expression into smaller terms using combinations, ultimately simplifying it into a polynomial.

Combinatorics

Combinatorics is the branch of mathematics that studies counting, arranging, and combination of objects. In binomial expansion, combinatorics help us determine the coefficients of each term. These coefficients are represented by \(\binom{n}{k}\), also known as binomial coefficients. They can be calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) denotes the factorial of \(n\). This concept is foundational in calculating the different ways the terms in the expansion combine, ensuring that all possible combinations are considered in the polynomial form.

Algebraic Expression

An algebraic expression involves numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In the context of the binomial expansion, the expression \((c+3)^5\) is an example of an algebraic expression. Each expanded term like \(c^5\), \(45c^4\), and \(3^5\) is part of the overall algebraic structure, showcasing how numbers and variables interact with each other through operations. Understanding how to manipulate and simplify algebraic expressions is crucial for solving complex problems in algebra.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial in its expanded form. A polynomial is a mathematical expression consisting of variables, coefficients, and operations of addition, subtraction, and multiplication. Using the Binomial Theorem, a binomial raised to a power is converted into its expanded polynomial form. In our exercise, \((c+3)^5\) is expanded into \(c^5 + 45c^4 + 90c^3 + 270c^2 + 405c + 243\). Each term results from applying both the binomial theorem and combinatorial methods, allowing us to view the expression in a detailed, expanded format.