Problem 51 - College Algebra with Modeling and Visualization

Question

Find the specified term. The sixth term of \((3 x-2 y)^{6}\)

Step-by-Step Solution

Answer

The sixth term is \(-576xy^5\).

Step 1: Identify the formula for the binomial expansion

To find a specific term in the expansion of a binomial expression \((a + b)^n\), we use the Binomial Theorem which states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).

Step 2: Determine the structure of the specific term

The \(r\)-th term in the binomial expansion is given by \(T_{r} = \binom{n}{r-1} a^{n-(r-1)} b^{r-1}\). In this problem, we need the 6th term, so \(r = 6\), \(a = 3x\), \(b = -2y\), and \(n = 6\).

Step 3: Calculate the binomial coefficient

For the 6th term, the binomial coefficient is \(\binom{6}{5}\). Calculate this using the formula for combinations: \(\binom{6}{5} = \frac{6!}{5!(6-5)!} = 6\).

Step 4: Find the powers of the components

Based on the general term formula \(T_{6} = \binom{6}{5} (3x)^{6-5} (-2y)^{5}\), calculate the powers: \((3x)^{1} = 3x\) and \((-2y)^{5} = (-32y^5)\).

Step 5: Calculate the 6th term

Substitute the values into the equation for \(T_{6}\): \(T_{6} = 6 \times 3x \times (-32y^5)\). Simplify the expression: \(T_{6} = 6 \times 3 \times (-32) \cdot x \cdot y^5\).

Step 6: Simplify the expression

Calculate the numerical part: \(6 \times 3 \times (-32) = -576\). Therefore, \(T_{6} = -576xy^5\).

Key Concepts

Binomial Expansion

Binomial Expansion is a mathematical method used to expand expressions that are raised to a power. It is essential when dealing with expressions of the form \((a + b)^n\). Instead of calculating each term by brute force multiplication, we use the Binomial Theorem to simplify the process.

In essence, binomial expansion allows us to write the polynomial form of a binomial expression.
Here's what happens:

• Start with the binomial expression like \((3x - 2y)^6\).
• Using the theorem, express it as a sum of terms with coefficients based on combinations.

The expanded form will include terms like \(a^{n-k}b^k\) where each term is structured according to its position in the sequence.
This method not only saves time but also reduces errors when expanding to higher powers.

Combination Formula

The Combination Formula is vital when dealing with binomial expansions as it determines the coefficients of each term. This formula is noted as \(\binom{n}{k}\) and read as "n choose k."

Here’s how it works:

• Use the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• The "!" denotes factorial, which is the product of all positive integers up to that number.

For example, to find the coefficient of the 6th term in \((3x-2y)^6\), we calculate \(\binom{6}{5} = 6\).
This tells us that each term's coefficient depends on its position and the total number of terms in the expansion. Using the combination formula is crucial for computing the correct coefficients quickly and accurately.

Powers of Binomial Terms

Understanding the powers in each term of a binomial expansion is crucial for solving problems like the one at hand. Each expanded term has components raised to specific powers based on their position.

Here's the breakdown:

• The term structure: \(a^{n-k}b^k\) helps calculate the correct powers for each component.
• Each term's components come from the original binomial, both being raised to complementary powers that add up to \(n\).

Take for instance the 6th term in \((3x-2y)^6\), the component \(3x\) is raised to the power \(6-5\), while \(-2y\) is raised to 5.
This method is essential as it dictates how the components combine, eventually leading to the full expanded expression.
In summary, accurate calculation of each term's powers is a critical step in the binomial expansion process.