Problem 4 - Discrete Mathematics with Applications

Question

Find the coefficient of each. \(x^{4} y^{5}\) in the expansion of \((2 x-3 y)^{9}\)

Step-by-Step Solution

Answer

The coefficient of \(x^4 y^5\) in the expansion of \((2x - 3y)^9\) is \(-489888\).

Step 1: Identify the term with \(x^4 y^5\)

In the binomial expansion, the term of interest will have the form: \(\binom{9}{k} (2x)^{9-k} (-3y)^{k}\) To find the value of \(k\) that produces \(x^4 y^5\), we equate the powers of \(x\) and \(y\): \(9 - k = 4 \Rightarrow k = 5\) and \(k = 5\) So, the term of interest in the binomial expansion is: \(\binom{9}{5} (2x)^{4} (-3y)^{5}\)

Step 2: Calculate the binomial coefficient

We need to compute the binomial coefficient: \(\binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2} = 126 \)

Step 3: Calculate the term with \(x^4 y^5\)

Now, we can substitute the binomial coefficient into the term of interest and find the coefficient of \(x^4 y^5\): \(126 \cdot (2x)^{4} \cdot (-3y)^{5}\) The coefficient of \(x^4 y^5\) in this term is: \(126 \cdot 2^{4} \cdot (-3)^{5} = 126 \cdot 16 \cdot (-243) = -489888\) Therefore, the coefficient of \(x^4 y^5\) in the expansion of \((2x - 3y)^9\) is \(-489888\).

Key Concepts

Binomial Coefficient

The binomial coefficient is a fundamental component in combinatorial mathematics and is used to determine the number of ways to choose a subset of elements from a larger set, without regard to the order of selection. In the context of binomial expansion, it helps in finding specific terms in polynomial expressions. The binomial coefficient is represented as \( \binom{n}{k} \), which is read as "n choose k," and is calculated using the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Here, \(n!\) ("n factorial") is the product of all positive integers up to \(n\), and \(k!\) is the product of all positive integers up to \(k\). The binomial coefficient is crucial in determining the number of ways a term can appear in the expansion of a binomial raised to a power.

Expansion of Polynomials

Expanding polynomials involves rewriting a polynomial expression in a longer, often more explicit form. When dealing with a binomial expression like \((2x - 3y)^9\), the expansion is achieved through the Binomial Theorem. The theorem allows this binomial to be expressed as a sum of terms, each consisting of a product of a binomial coefficient, a power of the first term in the binomial, and a power of the second term.

• The Binomial Theorem states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• In our specific example, the expansion reflects terms of the form: \(\binom{9}{k} (2x)^{9-k} (-3y)^k\)

By determining each term in the expansion, we gain insight into the distribution of terms with specific powers of variables, such as finding \(x^4 y^5\).

Coefficient Calculation

The calculation of coefficients in a polynomial expansion is a multi-step process that involves several calculations. Once you have identified the correct term using the binomial coefficient and powers of the variables, the next step is calculating the actual coefficient that accompanies the variables in that term.

• First, use the derived binomial coefficient, \( \binom{9}{5} = 126 \), which indicates the number of ways the term can occur.
• Next, calculate the contribution from each component of the term: \((2x)^4\) and \((-3y)^5\).
• \( (2x)^4 = 16x^4 \)
• \( (-3y)^5 = -243y^5 \)

Combine these calculations: \(126 \times 16 \times -243 = -489888\). The resulting coefficient for the term \(x^4 y^5\) is \(-489888\). This process illustrates how numerical coefficients are derived from the binomial expansion using algebraic manipulation.