Problem 18 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The expanded form of \((3x - 2y)^4\) is \(81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form \((a + b)^n\) into a sum involving terms of the form \(a^k b^{n-k}\). The general expansion formula is \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).

Step 2: Identify Components

In the binomial \((3x - 2y)^4\), identify \(a = 3x\), \(b = -2y\), and \(n = 4\). The goal is to expand using these components.

Step 3: Compute Binomial Coefficients

Use the binomial coefficient formula \(\binom{n}{k}\), where \(k\) ranges from 0 to \(n\). Compute each coefficient for \(n = 4\):- \(\binom{4}{0} = 1\)- \(\binom{4}{1} = 4\)- \(\binom{4}{2} = 6\)- \(\binom{4}{3} = 4\)- \(\binom{4}{4} = 1\)

Step 4: Expand the Expression

Apply the Binomial Theorem expansion:\[(3x - 2y)^4 = \sum_{k=0}^{4} \binom{4}{k} (3x)^{4-k} (-2y)^k\]This gives us:- For \(k = 0\), the term is \(1 \cdot (3x)^4 \cdot (-2y)^0 = 81x^4\)- For \(k = 1\), the term is \(4 \cdot (3x)^3 \cdot (-2y)^1 = -216x^3y\)- For \(k = 2\), the term is \(6 \cdot (3x)^2 \cdot (-2y)^2 = 216x^2y^2\)- For \(k = 3\), the term is \(4 \cdot (3x)^1 \cdot (-2y)^3 = -96xy^3\)- For \(k = 4\), the term is \(1 \cdot (3x)^0 \cdot (-2y)^4 = 16y^4\)

Step 5: Write the Final Expanded Expression

Combine all terms to write the final expanded form:\[(3x - 2y)^4 = 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4\]

Key Concepts

Binomial coefficient

The binomial coefficient is a crucial part of the binomial theorem, used to determine the coefficients in the expansion of a binomial expression. When you see a binomial raised to a power, as in \((a + b)^n\), the binomial coefficients are found using the combination formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n\) is the power and \(k\) is a specific term in the series.

Here's how you can break it down:

• \(n!\) ("n factorial") is the product of all positive integers up to \(n\).
• The formula \(\binom{n}{k}\) tells you how many ways you can pick \(k\) elements from a set of \(n\) elements.

These coefficients allow for the systematic expansion of \((a + b)^n\) into individual terms. For our exercise with \((3x - 2y)^4\), finding these coefficients allowed us to compute specific terms in the expansion.

Polynomial expansion

Polynomial expansion involves writing a binomial expression raised to a power as a polynomial. This means expressing it as a sum of multiple terms.

The binomial theorem guides the expansion process by using the formula for each term:\(a^{n-k}b^k\) multiplied by its corresponding binomial coefficient, \(\binom{n}{k}\).

When expanding \((3x - 2y)^4\), you break the task:

• Compute each term by substituting \(a = 3x\), \(b = -2y\), and \(n = 4\).
• Calculate the specific power for each term according to the decreasing powers of \(a\) and increasing powers of \(b\).
• Adjust the signs of each term if necessary, based on the term's part in the binomial.

Finally, sum up all the terms to achieve the expanded expression. This is known as polynomial expansion and is a methodical way to transform binomials into expanded algebraic expressions.

Algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operators (like plus and minus signs). These expressions can represent real-world quantities and allow complex problem solving.

In our exercise, \((3x - 2y)^4\) is an algebraic expression where:

• \(3x\) and \(-2y\) are individual terms.
• The expression involves a subtraction and multiplication as key operations.
• It's raised to a power, meaning it needs expansion for simplification.

Algebraic expressions like this can be simplified or transformed using various algebraic rules and theorems. By applying the binomial theorem, we expanded the given expression into a polynomial containing terms such as \(81x^4\) and \(-216x^3y\).

Understanding these expansions enables handling more complex algebraic tasks by breaking them into simpler parts.