Problem 13 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Row in Pascal's Triangle

Since we want to expand \((2x - 3y)^3\), we need the 4th row of Pascal's Triangle (remember, the first row is considered the 0th). The 4th row is \([1, 3, 3, 1]\). These numbers are the binomial coefficients for the expansion.

Step 2: Set Up the Binomial Expansion Formula

The general binomial expansion for \((a + b)^n\) is given by: \[C(n, 0) a^n b^0 + C(n, 1) a^{n-1} b^1 + C(n, 2) a^{n-2} b^2 + \, ... \, + C(n, n) a^0 b^n\] Using the coefficients from Pascal's Triangle: \(C(3,0), C(3,1), C(3,2), C(3,3)\), and setting \(a = 2x\) and \(b = -3y\), we will apply these to our expression.

Step 3: Compute Each Term Using Coefficients

Substitute the terms into the binomial expansion:1. First term: \(1 \cdot (2x)^3 \cdot (-3y)^0 = 1 \cdot 8x^3 \cdot 1 = 8x^3\)2. Second term: \(3 \cdot (2x)^2 \cdot (-3y)^1 = 3 \cdot 4x^2 \cdot (-3y) = -36x^2y\)3. Third term: \(3 \cdot (2x)^1 \cdot (-3y)^2 = 3 \cdot 2x \cdot 9y^2 = 54xy^2\)4. Fourth term: \(1 \cdot (2x)^0 \cdot (-3y)^3 = 1 \cdot 1 \cdot (-27y^3) = -27y^3\)

Step 4: Combine All Terms for the Final Expansion

Add all the terms from Step 3 together to form the full expanded expression:\[8x^3 - 36x^2y + 54xy^2 - 27y^3\] This is the expanded form of \((2x - 3y)^3\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful technique used to expand expressions raised to a power in the form \((a + b)^n\). This concept allows you to break down complex polynomial expressions into simpler terms.

• In our example, the expression \( (2x - 3y)^3 \) can be expanded using binomial expansion.
• The idea is to use the binomial theorem to distribute the terms systematically.

The formula for binomial expansion is:\[C(n, 0) a^n b^0 + C(n, 1) a^{n-1} b^1 + C(n, 2) a^{n-2} b^2 + \ldots + C(n, n) a^0 b^n\]To apply this formula, we use binomial coefficients which are derived from Pascal's Triangle. This method provides a step-by-step approach to expanding binomials, making it accessible for students to grasp complex calculations by breaking them down into manageable steps.

Binomial Coefficients

Binomial coefficients play a key role in binomial expansion. These coefficients are numbers that appear in Pascal's Triangle and are used to determine the multiplier for each term in the expansion.

• In our specific example \((2x - 3y)^3\), the binomial coefficients are \([1, 3, 3, 1]\) from the 4th row of Pascal's Triangle.
• These coefficients align with the powers of the expansion: they dictate how many times each term is multiplied in the expansion.

The coefficients are denoted as \(C(n, k)\), where \(n\) is the number of terms (or degree of the polynomial), and \(k\) is the specific term you're solving for. They can be calculated using the formula:\[C(n, k) = \frac{n!}{k!(n-k)!}\]With this foundational knowledge, binomial coefficients allow for precise, systematic polynomial expansion, ensuring accuracy and consistency in solving binomial expressions.

Polynomial Expansion

Polynomial expansion relates to the transformation of an expression like \((2x - 3y)^3\) into a series of terms. Each of these terms is a product of the original variables raised to various powers.

• Each term is calculated using the binomial coefficients identified in the binomial expansion process.
• For \((2x - 3y)^3\), the expanded form is \(8x^3 - 36x^2y + 54xy^2 - 27y^3\).

This transformation allows a more straightforward analysis and application of the polynomial since each part is explicitly simplified. Understanding how to perform a polynomial expansion enables students to handle high-power polynomial expressions with ease, breaking them into tangible parts and solving complex mathematical problems efficiently.