Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (4 x-3 y)^{5} $$

Step-by-Step Solution

Answer

The expanded form is \( 1024x^5 - 3840x^4y + 5760x^3y^2 - 4320x^2y^3 + 1620xy^4 - 243y^5 \).

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand binomials raised to a power. It states that \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \), where \( \binom{n}{k} \) is a binomial coefficient calculated as \( \frac{n!}{k!(n-k)!} \).

Step 2: Identify Components

In the expression \( (4x - 3y)^5 \), identify \( a = 4x \), \( b = -3y \), and \( n = 5 \). These will be used to apply the Binomial Theorem.

Step 3: Calculate Binomial Coefficients

List the binomial coefficients for \( n = 5 \): \( \binom{5}{0} = 1 \), \( \binom{5}{1} = 5 \), \( \binom{5}{2} = 10 \), \( \binom{5}{3} = 10 \), \( \binom{5}{4} = 5 \), \( \binom{5}{5} = 1 \).

Step 4: Expand Using the Binomial Theorem

Apply the theorem: \( (4x - 3y)^5 = \sum_{k=0}^{5} \binom{5}{k} (4x)^{5-k} (-3y)^k \). - For \( k = 0 \): \( \binom{5}{0}(4x)^5(-3y)^0 = 1 \times 1024x^5 = 1024x^5 \) - For \( k = 1 \): \( \binom{5}{1} (4x)^4 (-3y)^1 = 5 \times 256x^4 \times (-3y) = -3840x^4y \) - For \( k = 2 \): \( \binom{5}{2} (4x)^3 (-3y)^2 = 10 \times 64x^3 \times 9y^2 = 5760x^3y^2 \) - For \( k = 3 \): \( \binom{5}{3} (4x)^2 (-3y)^3 = 10 \times 16x^2 \times (-27y^3) = -4320x^2y^3 \) - For \( k = 4 \): \( \binom{5}{4} (4x)^1 (-3y)^4 = 5 \times 4x \times 81y^4 = 1620xy^4 \) - For \( k = 5 \): \( \binom{5}{5} (4x)^0 (-3y)^5 = 1 \times 1 \times (-243y^5) = -243y^5 \)

Step 5: Combine and Simplify

Add up the terms from each calculation:\[ 1024x^5 - 3840x^4y + 5760x^3y^2 - 4320x^2y^3 + 1620xy^4 - 243y^5 \]. This is the expanded form of the binomial expression.

Key Concepts

Binomial Expansion

When we come across expressions where two terms are raised to a power, the Binomial Theorem becomes our best friend. The binomial expansion takes an expression of the form \((a + b)^n\) and expands it into a sum of terms using the theorem. This method saves you from having to manually multiply the binomial by itself repeatedly, which can be quite laborious when \(n\), the exponent, is large.

The theorem states that \((a + b)^n\) can be expanded as a sum:

• \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

This means you'll calculate each term by determining the powers of \(a\) and \(b\), as well as the binomial coefficients. For example, the binomial \((4x - 3y)^5\) expanded using this theorem results in a polynomial where each term is a product of powers of \(4x\) and \(-3y\) multiplied by a specific coefficient.

Binomial Coefficients

Central to the binomial expansion are the binomial coefficients, denoted as \(\binom{n}{k}\), also known as 'combinations'. They are numbers that give us the weight of each term in the expansion and are essential in compiling the full expression. To calculate a binomial coefficient for specific values of \(n\) and \(k\), the formula is:

• \( \frac{n!}{k!(n-k)!} \)

Where \(!\) denotes a factorial, the product of all positive integers up to that number.

In practice, for the expression \((4x - 3y)^5\), you calculate the coefficients \(\binom{5}{0}\) through \(\binom{5}{5}\). These are the combinations of 5 items taken \(k\) at a time, equating to the sequences: 1, 5, 10, 10, 5, and 1.

Algebraic Expressions

Algebraic expressions like the one we expand using the Binomial Theorem consist of variables (like \(x\) and \(y\)), constants, and arithmetic operations. Mastering these is crucial for understanding how to manipulate and simplify them in broader mathematical contexts.

When dealing with the expansion of \((4x - 3y)^5\), each term involves calculating powers of \(4x\) and \(-3y\), and carefully multiplying by the binomial coefficient \(\binom{n}{k}\). Each of these elements highlights the importance of accurate algebraic manipulation. After expansion, you've transformed the original compact expression into a more extensive polynomial:

• \(1024x^5 - 3840x^4y + 5760x^3y^2 - 4320x^2y^3 + 1620xy^4 - 243y^5\).

This polynomial showcases how various algebraic operations combine, illustrating the richness and versatility of algebraic expressions.