Problem 19 - College Algebra

Question

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

Step-by-Step Solution

Answer

\((4x - 3y)^5 = 1024x^5 - 3840x^4y + 8640x^3y^2 - 8640x^2y^3 + 4860xy^4 - 243y^5\).

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 \(\binom{n}{k} a^{n-k} b^k\), where \(n\) is a non-negative integer and \(\binom{n}{k}\) is a binomial coefficient. For this problem, we have \((4x - 3y)^5\), which can be rewritten in the form \((a + b)^5\) by letting \(a = 4x\) and \(b = -3y\).

Step 2: Write the General Term Formula

We use the formula for the general term in the expansion: \(T_k = \binom{5}{k} (4x)^{5-k} (-3y)^k\), where \(k\) ranges from 0 to 5. This will give us each term in the expansion.

Step 3: Calculate the Binomial Coefficients

Compute the binomial coefficients \(\binom{5}{k}\) for \(k = 0, 1, 2, 3, 4, 5\). These are:- \(\binom{5}{0} = 1\)- \(\binom{5}{1} = 5\)- \(\binom{5}{2} = 10\)- \(\binom{5}{3} = 10\)- \(\binom{5}{4} = 5\)- \(\binom{5}{5} = 1\)

Step 4: Calculate and Simplify Each Term

Substitute \(k = 0, 1, 2, 3, 4, 5\) into the general term formula \(T_k = \binom{5}{k} (4x)^{5-k} (-3y)^k\), and simplify each:- For \(k = 0\): \(1 \cdot (4x)^5 \cdot 1 = 1024x^5\)- For \(k = 1\): \(5 \cdot (4x)^4 \cdot (-3y) = -3840x^4y\)- For \(k = 2\): \(10 \cdot (4x)^3 \cdot (9y^2) = 8640x^3y^2\)- For \(k = 3\): \(10 \cdot (4x)^2 \cdot (-27y^3) = -8640x^2y^3\)- For \(k = 4\): \(5 \cdot (4x) \cdot (81y^4) = 4860xy^4\)- For \(k = 5\): \(1 \cdot 1 \cdot (-243y^5) = -243y^5\)

Step 5: Combine All Terms

Combine the terms obtained in the previous step to form the binomial expansion:\[1024x^5 - 3840x^4y + 8640x^3y^2 - 8640x^2y^3 + 4860xy^4 - 243y^5\]

Key Concepts

Binomial Expansion

The Binomial Theorem is a powerful tool in algebra that simplifies the process of expanding expressions raised to a power. When you encounter a binomial expression such as \((4x - 3y)^5\), using the theorem helps us to quickly expand it into a series of terms. This method relies on the formula \((a + b)^n\), allowing us to escape tedious manual multiplication. Each component of the expression contributes to the final expanded form depending on its placement of powers. Often, the Binomial Expansion is expressed as:

• The sum of multiple terms based on the power \(n\).
• Each term powered respectively by \((a)\) and \((b)\).
• Use integer coefficients called binomial coefficients.

Applying the Binomial Theorem allows us to convert a complex expression into a manageable list of terms.

Binomial Coefficients

Binomial coefficients play a key role within the Binomial Theorem. These coefficients, denoted by \(\binom{n}{k}\), represent the number of ways to choose \(k\) items from a set of \(n\) items without regard to order. They create the 'weights' for each term in the binomial expansion. For our example \((4x - 3y)^5\), the coefficients determine the magnitude of each distinct product of powers in the expanded polynomial. To calculate these coefficients efficiently, Pascal's Triangle or directly using combinations formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) can be used. For \((4x-3y)^5\), each coefficient is:

• \(\binom{5}{0} = 1\)
• \(\binom{5}{1} = 5\)
• \(\binom{5}{2} = 10\)
• \(\binom{5}{3} = 10\)
• \(\binom{5}{4} = 5\)
• \(\binom{5}{5} = 1\)

The coefficients ensure that each term accounts for its full algebraic contribution to the expansion.

Algebraic Expressions

Algebraic expressions are statements composed of variables, constants, and arithmetic operations. In the exercise, the binomial expression consists of two terms \((4x - 3y)\) raised to the power of 5. Breaking these expressions according to rules of algebra is crucial, especially when dealing with expansions. Each term in the expansion results from applying powers to both alterative terms \((a)\) and \((b)\) respectively across different coefficients and powers of \(x\) and \(y\). These specifics form new algebraic expressions like \(1024x^5\) and \(-3840x^4y\). Recognizing and manipulating these expressions is essential for simplifying problems and obtaining precise results in multi-term expansions. Thus, understanding how each element affects the final expression helps streamline problem-solving in algebra.