Problem 13 - Precalculus: Mathematics for Calculus

Question

Pascal's Triangle Use Pascal's triangle to expand the expression. $$(2 x-3 y)^{3}$$

Step-by-Step Solution

Answer

The expression expands to \(8x^3 - 36x^2y + 54xy^2 - 27y^3\).

Step 1: Understanding Pascal's Triangle

Pascal's Triangle is a triangular array where each number is the sum of the two numbers directly above it. The row number corresponds to the power of the binomial. For the expansion of \((a + b)^3\), we need the 4th row (starting count from 0) of Pascal's Triangle, which is \([1, 3, 3, 1]\).

Step 2: Identifying the Binomial Components

The binomial expression given is \((2x - 3y)^3\). Here, \(a = 2x\) and \(b = -3y\). We will use these components in the expansion along with the coefficients from Pascal's Triangle.

Step 3: Applying Binomial Expansion Formula

The binomial expansion formula is \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Using the coefficients from Pascal's Triangle (1, 3, 3, 1), the expansion of \((2x - 3y)^3\) is:\(1 \cdot (2x)^3 \cdot (-3y)^0 + 3 \cdot (2x)^2 \cdot (-3y)^1 + 3 \cdot (2x)^1 \cdot (-3y)^2 + 1 \cdot (2x)^0 \cdot (-3y)^3\).

Step 4: Calculating Each Term

Calculate each term individually:- First term: \(1 \cdot (2x)^3 \cdot 1 = 8x^3\)- Second term: \(3 \cdot (2x)^2 \cdot (-3y) = 3 \cdot 4x^2 \cdot (-3y) = -36x^2y\)- Third term: \(3 \cdot (2x) \cdot (-3y)^2 = 3 \cdot 2x \cdot 9y^2 = 54xy^2\)- Fourth term: \(1 \cdot 1 \cdot (-3y)^3 = -27y^3\).

Step 5: Writing the Final Expanded Expression

Finally, combine the terms to write the full expanded expression: \(8x^3 - 36x^2y + 54xy^2 - 27y^3\).

Key Concepts

Binomial Expansion

Binomial expansion is a way to express powers of binomials in an expanded form. For any binomial expression \((a + b)^n\), the expansion leads to a sum of terms, each term being a product of a coefficient, a power of \(a\), and a power of \(b\). This expansion is made easier with the use of Pascal’s Triangle, as it provides the coefficients for each term.
The expansion follows a pattern that is consistent and predictable:

• The number of terms in the expansion is \(n+1\), where \(n\) is the power the binomial is raised to.
• Each term in the expansion represents a specific arrangement of items that arise from multiplying \((a + b)\) by itself \(n\) times.
• The coefficients of the terms correspond to the numbers in Pascal’s Triangle.

Understanding this concept is key as it simplifies the complex calculations involved in expanding binomials, making it much more accessible.

Binomial Theorem

The binomial theorem provides a formulaic way to expand expressions that are raised to a power, specifically of the form \((a + b)^n\). It establishes a relationship between the powers of the terms and the coefficients that arise from this expansion using combinations.
The formula for the binomial theorem is given by:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Where:

• \(n\) is the degree, or the power to which the binomial is raised.
• \(\binom{n}{k}\) is a binomial coefficient, representing the number of ways to choose \(k\) elements from a set of \(n\) elements, also found in Pascal's Triangle.
• \(a^{n-k}\) and \(b^k\) indicate the decreasing and increasing powers of \(a\) and \(b\) respectively across the terms.

This theorem is not only a foundation in algebra but also finds applications in calculus and probability, especially when dealing with polynomial functions.

Power of Binomial

The power of a binomial refers to raising a binomial expression to a specific exponent, as seen in cases like \((a+b)^3\). Understanding the power of binomials is essential when calculating probabilities, polynomial expansions, and even in ensuring the accuracy of numerical methods in calculus.
The power of the binomial determines:

• The number of terms in the expansion; for an \(n\)th power, there will be \(n+1\) terms.
• The degree of the resultant polynomial equal to the power of the binomial.
• The specific arrangement and coefficients of terms provided by the binomial theorem and Pascal's Triangle.

Exploring higher powers of binomials assists in developing intuitive comprehension of the fundamental principles of algebra and enhances problem-solving skills for more complex mathematical concepts.