Problem 4 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Required Row in Pascal's Triangle

To expand \((x-y)^5\), identify the row in Pascal's Triangle that corresponds to the power 5.The rows in Pascal's Triangle start with row 0. For power 5, use row 5, which is 1, 5, 10, 10, 5, 1.

Step 2: Assign Coefficients from Pascal's Triangle

The coefficients for the expansion of \((x-y)^5\) are determined by the numbers in the 5th row of Pascal's Triangle: 1, 5, 10, 10, 5, 1.

Step 3: Expand the Expression Using Binomial Theorem

Use the formula of binomial expansion which is:\[(x-y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} (-y)^k\]For \((x-y)^5\), substitute with the appropriate coefficients and terms:- First term: \(1 \cdot x^5\)- Second term: \(5 \cdot x^4 \cdot (-y)^1 = -5x^4y\)- Third term: \(10 \cdot x^3 \cdot (-y)^2 = 10x^3y^2\)- Fourth term: \(10 \cdot x^2 \cdot (-y)^3 = -10x^2y^3\)- Fifth term: \(5 \cdot x^1 \cdot (-y)^4 = 5xy^4\)- Sixth term: \(1 \cdot x^0 \cdot (-y)^5 = -y^5\).

Step 4: Combine the Results

Combine all the terms to get the expansion:\(x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a simple, yet powerful mathematical tool used extensively in combinatorics and probability. It is a triangular array of numbers where each number is the sum of the two numbers directly above it. Starting with the top of the triangle at row 0, each subsequent row builds upon the previous one. To construct Pascal's Triangle, begin with 1 at the top, and then for each new row, place 1 at the ends and fill in the interior numbers by adding the two numbers above them from the previous row.

For example, the first few rows are:

• Row 0: 1
• Row 1: 1, 1
• Row 2: 1, 2, 1
• Row 3: 1, 3, 3, 1
• Row 4: 1, 4, 6, 4, 1
• Row 5: 1, 5, 10, 10, 5, 1

Pascal's Triangle is particularly useful for finding the coefficients needed in binomial expansions.

Binomial Expansion

The Binomial Expansion is a method used to expand expressions raised to any given power, such as \((x-y)^5\). It employs the Binomial Theorem, which states that any binomial raised to the power of \(n\) can be expanded into a sum of terms in the form:

• \(\binom{n}{k} x^{n-k} y^k\)

Use this formula with the coefficients derived from Pascal's Triangle to systematically expand a binomial.

In our exercise example for \((x-y)^5\), you simply take the coefficients from the 5th row of Pascal’s Triangle (1, 5, 10, 10, 5, 1) and pair them with the appropriate powers of \(x\) and \(y\). Thus, the expression unfolds as:

• First term: \(1 \, \cdot \, x^5\)
• Second term: \(5 \, \cdot \, x^4 \, \cdot \, (-y)^1 = -5x^4y\)
• Third term: \(10 \, \cdot \, x^3 \, \cdot \, (-y)^2 = 10x^3y^2\)
• Fourth term: \(10 \, \cdot \, x^2 \, \cdot \, (-y)^3 = -10x^2y^3\)
• Fifth term: \(5 \, \cdot \, x^1 \, \cdot \, (-y)^4 = 5xy^4\)
• Sixth term: \(1 \, \cdot \, x^0 \, \cdot \, (-y)^5 = -y^5\)

Coefficients

Coefficients play a crucial role in algebraic expressions and expansions, acting as multipliers for terms in a polynomial. In the context of the Binomial Theorem and expansions, coefficients are used to determine the weight of each term in the expansion. They can be derived from Pascal's Triangle, where they correspond to the \(n\)th row based on the power to which the binomial is raised.

When expanding \((x-y)^5\), the coefficients are derived from the 5th row of Pascal’s Triangle: 1, 5, 10, 10, 5, and 1. These numbers tell us how many of each term \((x \, \text{and} \, y)\) will contribute to the whole expansion:

• The first and last terms are weighted by the smallest coefficient, 1.
• The central terms, having larger coefficients like 10, contribute maximally to the expression's final value.
• Positive coefficients are used directly, while the sign must be considered, especially when terms are raised to an odd power, making their influence negative, as seen with terms involving \((-y)^k\).