Problem 8 - Precalculus: Mathematics for Calculus

Question

Pascal's Triangle Use Pascal's triangle to expand the expression. $$(x-y)^{5}$$

Step-by-Step Solution

Answer

The expansion is \(x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5\).

Step 1: Understanding Pascal's Triangle

Pascal's Triangle is a triangular array of binomial coefficients. Each row corresponds to the coefficients of the expanded form of a binomial power. The first row (row 0) is \([1]\). Row 1 is \([1, 1]\), and so on. To solve \((x-y)^5\), we need the 6th row (since we start from row 0).

Step 2: Finding Row 5 of Pascal's Triangle

The 5th row (index 5 corresponds to the power) of Pascal's Triangle is \([1, 5, 10, 10, 5, 1]\). These numbers are the binomial coefficients for each term in the expansion of \((x-y)^5\).

Step 3: Setting Up the Expression using Binomial Theorem

Using the binomial theorem, \((x-y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). Substitute \(n = 5\) and the coefficients from the 5th row: \(1x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5\).

Step 4: Combining the Terms

Combine the terms using the coefficients from Pascal's Triangle: \((x-y)^5 = x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5\).

Step 5: Final Review

Ensure each term in the expression corresponds correctly to the coefficients \(1, 5, 10, 10, 5, 1\) and that the powers of \(x\) and \(y\) decrease and increase appropriately. This completes the binomial expansion.

Key Concepts

Binomial Coefficients

Binomial coefficients are the numbers that appear in Pascal's Triangle, they play a crucial role in the expansion of binomials. Each coefficient in a row of Pascal's Triangle corresponds to a term in the expansion of \((x+y)^n\). These coefficients are denoted as \(\binom{n}{k}\), representing the number of ways to choose \(k\) elements from \(n\) elements without regard to order. In mathematical terms, this is expressed as:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(!\) denotes a factorial, the product of all positive integers up to a given number.
For example, the 5th row of Pascal's Triangle is \([1, 5, 10, 10, 5, 1]\). Each of these numbers is a binomial coefficient, used to expand a power such as \((x-y)^5\). These coefficients explain the pattern of how each term combines in the expansion, dictating how many times each variable appears in the multiplied terms.

Binomial Theorem

The Binomial Theorem provides us a powerful way to expand expressions of the form \((x+y)^n\) where \(n\) is a non-negative integer. This theorem states that:\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]In this expansion, \(\binom{n}{k}\) refers to the binomial coefficient and indicates how many times each term appears in the binomial expansion. It ensures that each term in the expansion is weighted correctly according to its position in the expression.
Using the binomial theorem, you can quickly identify the terms in the expansion without manually multiplying the expression multiple times. Instead, by using binomial coefficients from Pascal's Triangle and systematic changes in the powers of \(x\) and \(y\), you can expand any binomial raised to a power efficiently and accurately.

Binomial Expansion

The process of binomial expansion involves using the binomial theorem to spread out a binomial expression raised to a power, like \((x-y)^n\), into a sum of terms. Each term consists of a product of powers of the individual components of the binomial with corresponding coefficients from Pascal's Triangle.For example, with \((x-y)^5\), the binomial expansion follows:

• Start from the largest power of \(x\), which is \(x^5\), and the smallest power of \(y\), since the second term \(y\) is multiplied to zero initially.
• Lower the power of \(x\) by one and increase the power of \(y\) by one in each subsequent term.
• Apply the binomial coefficients \([1, 5, 10, 10, 5, 1]\) to each term to weight them correctly.

Thus, you achieve the expansion:\((x-y)^5 = x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5\).This systematic way of working through the terms reduces the risk of calculation error and ensures that each produced term accurately contributes to the final expression.