Question

In the expansion of \((3 x-2 y)^{18},\) what are the coefficients of: (a) \(x^{5} \cdot y^{13}\) (b) \(x^{3} \cdot y^{15}\)

Short answer

The coefficient for \(x^5 \cdot y^{13}\) is \(-17082518528\), and for \(x^3 \cdot y^{15}\) is \(-72259011328\).

Step-by-step solution

Understand the Binomial Theorem

The Binomial Theorem provides a formula to expand expressions in the form of \((a+b)^n\). The theorem states: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]. We can use this to determine the coefficients in the expansion of \((3x - 2y)^{18}\).

Identify components for the term x^5 * y^13

We need to express the term \(x^5 \cdot y^{13}\) in the binomial expansion format. Here, \(n = 18\), and for this term, we want \(b = -2y\) raised to the 13th power, and \(a = 3x\) raised to the 5th power. Therefore, \(k = 13\).

Apply the Binomial Coefficient for x^5 * y^13

The coefficient of the term can be found using the expression \(\binom{18}{13} (3x)^{18-13} (-2y)^{13}\). We calculate: \[\binom{18}{13} = \binom{18}{5}, \] because \(\binom{n}{k} = \binom{n}{n-k}\).\[(3)^5 = 243, \] and \[(-2)^{13} = -8192.\]

Compute the Coefficient for x^5 * y^13

Calculate \[\binom{18}{5} = 8568 \] (using Pascal's triangle or a calculator).The coefficient is then: \[8568 \times 243 \times (-8192)\]. Calculate this product to find the coefficient.

Identify components for the term x^3 * y^15

We need to express the term \(x^3 \cdot y^{15}\) in the binomial expansion format. Here, \(n = 18\), and for this term, we want \(b = -2y\) raised to the 15th power, and \(a = 3x\) raised to the 3rd power. Therefore, \(k = 15\).

Apply the Binomial Coefficient for x^3 * y^15

The coefficient of this term can be found using the expression \(\binom{18}{15} (3x)^{18-15} (-2y)^{15}\). We calculate: \[\binom{18}{15} = \binom{18}{3}, \] because \(\binom{n}{k} = \binom{n}{n-k}\).\[(3)^3 = 27, \] and \[(-2)^{15} = -32768.\]

Compute the Coefficient for x^3 * y^15

Calculate \[\binom{18}{3} = 816 \] (using Pascal's triangle or a calculator). The coefficient is then: \[816 \times 27 \times (-32768)\]. Calculate this product to find the coefficient.

Key concepts

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial raised to a power as a sum of terms. In the context of the Binomial Theorem, this means taking \((a + b)^n\) and writing it as a sum of terms that involve powers of both \(a\) and \(b\).

To expand a polynomial using the Binomial Theorem:

• Identify the basic components of the expression, namely \(a\) and \(b\), as well as the exponent \(n\).
• Apply the Binomial Theorem formula: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).
• This results in a series of terms that each have a different coefficient determined by a binomial coefficient.

This method allows the accurate calculation of specific terms in a polynomial, such as finding the coefficient of a term when needing a particular form like \(x^5\cdot y^{13}\). It makes working with polynomials simpler and allows us to find values and coefficients directly without expanding the whole expression.

Combinatorics

Combinatorics is a field of mathematics focused on counting combinations and permutations. It plays an essential role in the Binomial Theorem, particularly when calculating binomial coefficients.

In practical terms, combinatorics helps us determine how many ways we can pick \(k\) items out of a total of \(n\) without considering order. This is expressed as \(\binom{n}{k}\), which is read as "n choose k". It gives us the number of ways to choose \(k\) elements from a set of \(n\) elements.

Why is this important for polynomial expansion?

• The binomial coefficients indicate how many times each term of a particular form will appear in the expansion.
• Combinatorial calculations are crucial for determining the correct coefficients in complex expansions.
• The binomial coefficients reduce the necessity to perform redundant calculations by using known patterns from combinatorics.

For students, understanding the relationship between combinatorics and polynomial expansions is vital. It simplifies the calculation of coefficients drastically and provides an organized method to solve seemingly complex problems.

Binomial Coefficients

Binomial coefficients are the central component of the Binomial Theorem, represented as \(\binom{n}{k}\). They indicate how each term in the expansion of \((a+b)^n\) will contribute to the total sum when considering a specific term.

To calculate a binomial coefficient:

• Use the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• "!" denotes factorial, meaning \(n! = n \times (n-1) \times (n-2) \times...\times 1\).
• These calculations are often simplified using Pascal's Triangle, where each number is the sum of the two directly above it.

For example, to find \(\binom{18}{5}\):

• Using the formula, it results in \(\frac{18!}{5!(18-5)!}\).
• Or you could look it up in Pascal’s Triangle quickly, which yields 8586.

Understanding the role of binomial coefficients enables solving for particular terms without full expansion. It crafts a clear picture of the pyramid structure of polynomial terms and improves computational efficiency in algebraic calculations.