Problem 9 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

\((2x - 3y)^3 = 8x^3 - 36x^2y + 54xy^2 - 27y^3\)."

Step 1: Identify the Binomial Coefficients

Pascal's Triangle helps us find the coefficients for the binomial expansion. For \((a+b)^n\), the coefficients are found in the \(n^{th}\) row of Pascal's Triangle. Since our expression is \((2x - 3y)^3\), we need the 3rd row: \(1, 3, 3, 1\).

Step 2: Apply the Binomial Theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = 2x\), \(b = -3y\), and \(n = 3\).

Step 3: Compute Each Term of the Expansion

Using the coefficients \(1, 3, 3, 1\) and substituting \(a = 2x\) and \(b = -3y\) into the formula, we expand:1. First term: \(1 \cdot (2x)^3 \cdot (-3y)^0 = 8x^3\)2. Second term: \(3 \cdot (2x)^2 \cdot (-3y)^1 = -36x^2y\)3. Third term: \(3 \cdot (2x)^1 \cdot (-3y)^2 = 54xy^2\)4. Fourth term: \(1 \cdot (2x)^0 \cdot (-3y)^3 = -27y^3\).

Step 4: Compile the Final Expanded Expression

Combine the terms calculated:\((2x - 3y)^3 = 8x^3 - 36x^2y + 54xy^2 - 27y^3\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that provides a way to expand expressions raised to a power. It applies to expressions of the form \((a + b)^n\). When you have a binomial raised to a power, instead of multiplying it out directly, the binomial theorem gives you a formula to find the expanded expression quickly.

The theorem states that:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, \( \binom{n}{k} \) represents the binomial coefficient and each term in the expansion is the product of a coefficient, a power of \(a\), and a power of \(b\).

• \(a\) and \(b\): The components of the binomial.
• \(n\): The exponent indicating how many times the binomial is multiplied by itself.
• Binomial Coefficients: These are identified in the expansion process, often using Pascal's Triangle.

The key to applying the Binomial Theorem is being able to identify and use these coefficients properly in your calculations.

Binomial Coefficients

Binomial coefficients are a special set of numbers that appear in the expansion of a binomial expression. They are denoted as \( \binom{n}{k} \), read as "n choose k," and calculate the number of ways to choose \(k\) elements from \(n\) elements. This concept is crucial in the context of binomial expansions.

The coefficients can be found using Pascal's Triangle, where each row represents the coefficients for a particular power of a binomial. For example, to expand \((a + b)^n\), you would look for the \(n^{th}\) row in the triangle.

• Symmetric: The rows of Pascal's Triangle are symmetric, which means that \( \binom{n}{k} = \binom{n}{n-k} \).
• Recursive Formula: The coefficients follow the rule \( \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} \).
• Sum of Powers: The sum of the numbers in the \(n^{th}\) row equals \(2^n\).

If you're comfortable with these concepts, using binomial coefficients to expand expressions becomes much more accessible.

Polynomial Expansion

Polynomial expansion involves breaking down an expression into a sum of simpler terms. When working with binomials raised to a power, the expansion is achieved through the use of the binomial theorem. The process translates a polynomial like \((a + b)^n\) into a series of terms that are easier to work with and understand.

The expansion adheres to the pattern dictated by the binomial theorem, using the binomial coefficients and the powers of the terms \(a\) and \(b\). Each term in the polynomial expansion has three key parts:

• Binomial Coefficient: Gives the weight of each term based on its position in the expansion.
• Powers of \(a\): Decrease from \(n\) to 0 as you move through the terms.
• Powers of \(b\): Increase from 0 to \(n\).

Through polynomial expansion, complex algebraic expressions become more manageable, allowing for further analysis or simplification in mathematical computations.