Problem 46 - Beginning and Intermediate Algebra

Question

Use the binomial theorem to expand each expression. $$\left(w^{3}+2\right)^{3}$$

Step-by-Step Solution

Answer

The expanded expression using the binomial theorem is \(\left(w^{3}+2\right)^{3}= w^9 + 6w^6 + 12w^3 + 8\).

Step 1: Identify the terms and the exponent

We are given the expression \(\left(w^{3}+2\right)^{3}\), so we can identify \(a = w^3\) , \(b = 2\) and \(n = 3\).

Step 2: Apply the binomial theorem

Using the binomial theorem formula, we can expand the given expression as follows: \(\begin{aligned} \left(w^{3}+2\right)^{3} &= \sum_{k=0}^{3} \binom{3}{k} (w^3)^{3-k} (2)^k \\ \end{aligned}\)

Step 3: Calculate the binomial coefficients and simplify the terms

Now, we will calculate the binomial coefficients and simplify the terms for each k value in the sum: \(k = 0\): \(\binom{3}{0} = 1\), \((w^3)^{3-0} = (w^3)^3\), \((2)^0 = 1\) \(k = 1\): \(\binom{3}{1} = 3\), \((w^3)^{3-1} = (w^3)^2\), \((2)^1 = 2\) \(k = 2\): \(\binom{3}{2} = 3\), \((w^3)^{3-2} = w^3\), \((2)^2 = 4\) \(k = 3\): \(\binom{3}{3} = 1\), \((w^3)^{3-3} = 1\), \((2)^3 = 8\)

Step 4: Substitute the values into the sum and simplify

Now, we will substitute the values we obtained in Step 3 into the sum and simplify the expression: \(\begin{aligned} \left(w^{3}+2\right)^{3} &= \sum_{k=0}^{3} \binom{3}{k} (w^3)^{3-k} (2)^k \\ &= \binom{3}{0}(w^3)^{3}(2)^0 + \binom{3}{1}(w^3)^{2}(2)^1 + \binom{3}{2}(w^3)^{1}(2)^2 + \binom{3}{3}(w^3)^{0}(2)^3 \\ &= 1 \cdot (w^9) \cdot 1 + 3 \cdot (w^6) \cdot 2 + 3 \cdot (w^3) \cdot 4 + 1 \cdot 1 \cdot 8 \\ &= w^9 + 6w^6 + 12w^3 + 8 \end{aligned}\) The expanded expression is \(w^9 + 6w^6 + 12w^3 + 8\).

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of expressing a power of a binomial expression as a sum of terms. A binomial expression is simply an algebraic expression that contains two terms, such as \((x + y)\). When you raise a binomial to a power, you use the binomial theorem to expand it into a series. This theorem simplifies the process by providing a formula that gives each term in the expansion.

The general form of a binomial expansion is represented by the formula:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, \(n\) represents the power, \(a\) and \(b\) are the terms of the binomial, and \(\binom{n}{k}\) are the binomial coefficients. This method is useful because it breaks down a complex algebraic expression into simpler components that are easier to manage. You only need to calculate each term separately and then combine them to get the final expanded expression.

Binomial Coefficients

Binomial coefficients are an essential part of the binomial theorem. They are the numbers that occur as coefficients in the expansion of the binomial expression. These coefficients can be found using Pascal's Triangle, but they are also directly calculated using combinations in mathematics.

The binomial coefficient \(\binom{n}{k}\) is read as \(n\) choose \(k\) and is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]The "!" denotes a factorial, which is the product of all positive integers up to that number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

In the given exercise, when expanding \((w^3+2)^3\), the binomial coefficients are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\). These coefficients multiply the respective terms \((w^3)^{3-k}\) and \((2)^k\) in the expansion, leading to the final result of the expanded polynomial.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. They serve as a fundamental aspect of algebra and help model real-world situations in mathematical terms.

The exercise under review involves the algebraic expression \((w^3 + 2)^3\), where you have to expand it using the binomial theorem. Algebraic expressions can be as simple as a single variable \(x\) or more complex like the one mentioned above. In any form, these expressions are vital for solving equations and performing algebraic manipulations.

By understanding how to expand expressions like \(w^3 + 2\), students gain mastery over manipulating terms and coefficients to achieve desired results in calculations. These skills are not only theoretical but also applicable in various fields such as engineering, physics, computer science, and economics. Recognizing patterns and being able to expand and simplify expressions can greatly enhance problem-solving capabilities.