Problem 18 - College Algebra

Question

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (3 x-2 y)^{4} $$

Step-by-Step Solution

Answer

\((3x - 2y)^4 = 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4\)."

Step 1: Write out the Binomial Theorem

The Binomial Theorem is written as \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This will help us expand any binomial expression of the form \((a+b)^n\). For our problem, \(a = 3x\), \(b = -2y\), and \(n = 4\).

Step 2: Determine the Binomial Coefficients

Use the formula for binomial coefficients \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) for each term where \(k\) ranges from 0 to 4, so \(n = 4\). The coefficients are \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), and \(\binom{4}{4} = 1\).

Step 3: Expand Each Term of the Binomial

Using the binomial theorem, calculate each term for the binomial expansion:- For \(k = 0\): \(1 \cdot (3x)^4 \cdot (-2y)^0 = 81x^4\)- For \(k = 1\): \(4 \cdot (3x)^3 \cdot (-2y)^1 = -216x^3y\)- For \(k = 2\): \(6 \cdot (3x)^2 \cdot (-2y)^2 = 216x^2y^2\)- For \(k = 3\): \(4 \cdot (3x)^1 \cdot (-2y)^3 = -96xy^3\)- For \(k = 4\): \(1 \cdot (3x)^0 \cdot (-2y)^4 = 16y^4\)

Step 4: Write the Complete Expansion

Combine all the terms from Step 3 to get the full binomial expansion: \((3x - 2y)^4 = 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4\).

Key Concepts

Binomial Expansion

The binomial expansion is a way to express the power of a binomial as a sum of terms. If you have an expression like \((a + b)^n\), the Binomial Theorem allows us to expand it into a series of terms. Each term consists of a binomial coefficient, a power of \(a\), and a power of \(b\). It's very useful for simplifying expressions in algebra.

In our exercise, we expanded \((3x - 2y)^4\) using this very method. By doing so, we transformed a complex expression into a more manageable form, making it easier to work with.

Binomial Coefficients

Binomial coefficients are essential in determining each term's weight in a binomial expansion. They tell us how many ways we can choose elements from a set, which correlates to the number of times each product of \(a\) and \(b\) appears in the expansion.

These coefficients are calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

For our specific problem, we calculated the binomial coefficients for values of \(k\) from 0 to 4, arriving at the sequence: 1, 4, 6, 4, and 1. These coefficients correspond to the different terms in the expansion process.

Polynomials

A polynomial is simply an expression with multiple terms, where each variable has non-negative integer exponents. It's fundamental in algebra to handle these mathematical expressions gracefully.

The concept of a polynomial is crucial for understanding binomial expansion because after expanding a \((a + b)^n\), the resulting expression is essentially a polynomial.

In our exercise, once we expanded the binomial, we ended up with a polynomial of five terms: 81x^4, -216x^3y, 216x^2y^2, -96xy^3, and 16y^4.

Algebra

Algebra is a wide and foundational branch of mathematics focused on symbols and the rules for manipulating these symbols. It provides a clear and structured way to represent numbers and their relationships.

• Symbols such as \(x\) and \(y\) allow for the generalization of arithmetic.
• Through algebra, we can solve equations, expand expressions, and explore a wide range of mathematical principles.

In our exercise, we used algebraic principles to expand \((3x - 2y)^4\) into a series of terms, making it simpler to analyze or compute when needed. Algebra helps bridge the gap between pure arithmetic numbers and more abstract mathematical concepts.