Problem 28 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The expanded form is \(8 + 36x^2 + 54x^4 + 27x^6\).

Step 1: Understanding the Binomial Theorem

The binomial theorem provides a formula for expanding any expression of the form \((a+b)^n\). It states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = 2\), \(b = 3x^2\), and \(n = 3\).

Step 2: Apply the Binomial Coefficient Formula

For \(n = 3\), the binomial coefficients are calculated using \(\binom{n}{k}\). The coefficients are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).

Step 3: Calculate Each Term

Calculate each term of the expansion: - For \(k=0\): \(\binom{3}{0} \cdot 2^{3} \cdot (3x^2)^{0} = 1 \times 8 \times 1 = 8\)- For \(k=1\): \(\binom{3}{1} \cdot 2^{2} \cdot (3x^2)^{1} = 3 \times 4 \times 3x^2 = 36x^2\)- For \(k=2\): \(\binom{3}{2} \cdot 2^{1} \cdot (3x^2)^{2} = 3 \times 2 \times 9x^4 = 54x^4\)- For \(k=3\): \(\binom{3}{3} \cdot 2^{0} \cdot (3x^2)^{3} = 1 \times 1 \times 27x^6 = 27x^6\)

Step 4: Combine the Terms

Add the calculated terms together to obtain the expanded form: \(8 + 36x^2 + 54x^4 + 27x^6\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a key concept in algebra that allows us to take expressions raised to a power and express them as a sum of terms. This is often achieved using the binomial theorem when dealing with expressions of the form \((a+b)^n\).
In our exercise, we are expanding \((2 + 3x^2)^3\) using this theorem. The goal is to break down the expression into simpler individual terms where each term's power and coefficient can easily be calculated.
By expanding the binomial, we can easily handle and simplify complex algebraic expressions which are crucial for solving more advanced mathematical problems.

Binomial Coefficients

Binomial coefficients are numerical factors that appear as part of the polynomial terms when expanding a binomial expression using the binomial theorem. They are represented by \(\binom{n}{k}\), which denotes 'n choose k', referring to the number of ways to choose \(k\) elements from a total of \(n\) elements.
Here's how we calculated the coefficients for the expansion of \((2 + 3x^2)^3\):

• \(\binom{3}{0} = 1\)
• \(\binom{3}{1} = 3\)
• \(\binom{3}{2} = 3\)
• \(\binom{3}{3} = 1\)

These coefficients help in computing each term of the expansion by scaling them appropriately, ensuring that the expanded polynomial maintains the same overall structure and magnitude as the original expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition and multiplication) that represent a specific value. The expression \((2 + 3x^2)^3\) is an example of an algebraic expression where both constants and variables are combined in a composition.
Understanding how to manipulate these expressions through means like polynomial expansion is a fundamental skill in algebra. It allows us to simplify, rearrange, and solve equations that may originally appear complex.
By expanding the expression, we can observe the relationship between terms and easily understand the effects of each component (including coefficients like 2 and 3, and variable expressions like \(x^2\)) in the context of the entire polynomial.