Problem 20 - Algebra 2 and Trigonometry

Question

In \(18-20,\) for the given expansion, identify which term is shown and write the next term. $$ \left(\frac{x}{2}-2 y\right)^{11} ; \frac{165}{4} x^{7} y^{4} $$

Step-by-Step Solution

Answer

The current term is the 5th term (for \( k=4 \)), and the next term is \(-231x^6y^5\).

Step 1: Identify the Binomial Coefficient

In the given binomial expansion \( \left(\frac{x}{2} - 2y\right)^{11} \), the expansion follows the pattern given by the Binomial Theorem:\[ \sum_{k=0}^{n} \binom{n}{k} (a)^{n-k} (b)^k \]where \( a=\frac{x}{2} \), \( b = -2y \), and \( n=11 \).The given term is \( \frac{165}{4} x^7 y^4 \), which suggests that \( n-k=7 \) and \( k=4 \). First, check the binomial coefficient \( \binom{11}{4} \):\[ \binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330 \].

Step 2: Determine the Powers of Terms

Using the values \( n - k = 7 \), \( k = 4 \), verify the powers of \( a \) and \( b \):\[ (a)^{n-k} = \left( \frac{x}{2} \right)^7 = \frac{x^7}{2^7} = \frac{x^7}{128} \]\[ (b)^{k} = (-2y)^4 = 16y^4 \].

Step 3: Combine Terms and Compute

Now combine to check if the term matches:\[ 330 \times \frac{x^7}{128} \times 16y^4 \]Find the coefficient:\[ 330 \times \frac{16}{128} = \frac{330 \times 16}{128} = \frac{330}{8} = \frac{165}{4} \]This verifies the given coefficient of the term.

Step 4: Identify the Current Term and Find the Next Term

From the preceding steps, the placed term corresponds to \( k = 4 \). To find the next term, compute for \( k = 5 \):\[ \binom{11}{5} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462 \].

Step 5: Calculate Next Term

Calculate based on \( k = 5 \), so \( n-k = 6 \):\[ \left( \frac{x}{2} \right)^6 = \frac{x^6}{64} \]\[ (-2y)^5 = -32y^5 \]Combine them:\[ 462 \times \frac{x^6}{64} \times (-32y^5) \]\[ = 462 \times (-\frac{x^6}{2}y^5) \]\[ = -231x^6y^5 \].

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful formula for expanding expressions that are raised to a power. Imagine you have a binomial expression, like \((a - b)^n\), and you want to expand it without multiplying the expression by itself repeatedly. The Binomial Theorem makes this task simpler by giving you a formula to follow:

• It's denoted as: \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• Here, \(n\) is the power to which the binomial is raised.
• \(a\) and \(b\) are the two terms in the binomial expression.

This formula allows you to find each term in the expansion by varying \(k\) from 0 to \(n\). For each value of \(k\), you calculate a different term. The Binomial Theorem helps in quickly expanding expressions, especially useful in higher mathematics, statistics, and algebra.

Binomial Coefficient

The Binomial Coefficient plays an essential role in finding each term in a binomial expansion. It's represented by \(\binom{n}{k}\), often pronounced as "n choose k." This coefficient shows you how many different ways you can choose \(k\) elements from a total of \(n\) elements, without considering the order.

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

In the given exercise, we identified the binomial coefficient by matching the term provided in the problem to the coefficients calculated. Knowing how to compute this coefficient is crucial because it forms the basis of determining each term's contribution to the overall expansion.

Polynomial Expansion

Polynomial Expansion is about expressing a power of a binomial as a sum of terms. Each term in this sum is a product of a coefficient, and powers of the variables involved. When expanding, we convert expressions like \((x+y)^n\) into a polynomial form.

• Each term in the polynomial can be extracted using the Binomial Theorem.
• The powers in the expansion follow a pattern: one variable's power decreases, while the other increases.

This expansion technique is often used to simplify calculations in algebra and calculus. By converting a binomial into a polynomial, it becomes easier to integrate, differentiate, or solve within an equation. The example given involves finding both current and next terms of a polynomial expansion using this method. This understanding enables solving more complex equations and systems that appear frequently in higher mathematics.