Problem 38 - Algebra for College Students

Question

Expand each binomial and simplify.\(\left(a^{2}-3 b^{3}\right)^{4}\)

Step-by-Step Solution

Answer

\(a^8 - 12a^6b^3 + 54a^4b^6 - 108a^2b^9 + 81b^{12}\)

Step 1: Identify the Binomial Coefficients

When expanding the binomial \((a^2 - 3b^3)^4\), use the binomial theorem, which states \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). For the binomial \((a^2 - 3b^3)\), the coefficients are given by \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \binom{4}{4}\). These coefficients are 1, 4, 6, 4, and 1.

Step 2: Apply the Binomial Expansion

Expand the binomial \((a^2 - 3b^3)^4\) using the coefficients identified: \(\sum_{k=0}^{4} \binom{4}{k} (a^2)^{4-k}(-3b^3)^k\).

Step 3: Calculate Each Term

Substitute and simplify each term of the expansion:- For \(k=0\): \(\binom{4}{0} (a^2)^4 (-3b^3)^0 = 1 \cdot a^8 = a^8\) - For \(k=1\): \(\binom{4}{1} (a^2)^3 (-3b^3)^1 = 4 \cdot a^6 \cdot (-3b^3) = -12a^6b^3\)- For \(k=2\): \(\binom{4}{2} (a^2)^2 (-3b^3)^2 = 6 \cdot a^4 \cdot 9b^6 = 54a^4b^6\)- For \(k=3\): \(\binom{4}{3} (a^2)^1 (-3b^3)^3 = 4 \cdot a^2 \cdot (-27b^9) = -108a^2b^9\)- For \(k=4\): \(\binom{4}{4} (a^2)^0 (-3b^3)^4 = 1 \cdot 81b^{12} = 81b^{12}\)

Step 4: Combine the Terms

Combine the simplified terms to form the expanded expression:\[ a^8 - 12a^6b^3 + 54a^4b^6 - 108a^2b^9 + 81b^{12} \]

Key Concepts

Binomial Theorem

The Binomial Theorem provides a way to expand expressions that are raised to a power, like \( (x + y)^n \). This theorem is particularly handy in algebra because it allows us to break down complex polynomial expansions into manageable parts. The expression is expanded as:\

• \( (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \)

Here, \( \binom{n}{k} \) represents the binomial coefficients, which dictate how each term in the expansion is computed. These coefficients are determined by combinatorial mathematics and can be found using Pascal's Triangle. The Binomial Theorem helps us systematize the expansion process by providing a clear formula for each resultant term. We apply this equation when solving exercises like expanding \( (a^2 - 3b^3)^4 \), making the solution precise and methodical.

Polynomial Expansion

Exploring polynomial expansion involves expanding expressions of polynomials into simpler forms. Using the binomial theorem, a binomial expression \( (a^2 - 3b^3)^4 \) can be expanded into a series of terms. Each term in this series results from the application of the binomial theorem, bringing both individual variables and powers together.This process requires:

• Breaking down the binomial expression into several separate terms, each associated with a binomial coefficient.
• Combining powers of individual elements such as \( a^2 \) and \( b^3 \) based on their respective exponents.
• Simplifying each term by multiplying coefficients and applying the exponent laws.

Polynomial expansion is crucial in algebra as it helps in simplifying complex algebraic expressions, making them easier to evaluate or integrate further into mathematical problems.

Algebraic Expressions

Algebraic expressions serve as the foundation for many problems in algebra, making an understanding of them critical for manipulating and solving mathematical equations. These are expressions that consist of variables, constants, and operators arranged in a specific format to define quantities. For the expression \( (a^2 - 3b^3)^4 \), we identify that:

• \( a^2 \) and \( -3b^3 \) are the two terms involved in the binomial format.
• The powers and constants allow us to transform these expressions efficiently using algebraic rules such as distribution, exponentiation, and simplification.

Understanding how to manipulate these expressions through addition, subtraction, multiplication, and division can greatly aid in successfully applying more advanced algebraic topics like binomial expansion and polynomial addition.

Combinatorial Coefficients

Combinatorial coefficients, often referred to as binomial coefficients, play an essential role in binomial expansions. Represented by \( \binom{n}{k} \), these coefficients represent the number of ways to choose \( k \) elements from a set of \( n \) elements, demonstrating their combinatorial origins.
The computation of each binomial coefficient is crucial in expanding \( (x + y)^n \), as it determines the multiplier for each term in the expansion equation. The coefficients for an expression like \( (a^2 - 3b^3)^4 \) are computed and identified as 1, 4, 6, 4, and 1, which guide the final multiplied results to derive the polynomial's expanded form.
This concept not only simplifies the calculation of expansions but also connects algebra with combinatorial mathematics, underlying the importance of understanding mathematical relationships across different topics.