Problem 37 - Algebra 2

Question

Expand each binomial. $$ (2 x-2 y)^{6} $$

Step-by-Step Solution

Answer

The expanded form of \( (2x - 2y)^6 \) is \(64x^6 - 384x^5y + 960x^4y^2 - 1280x^3y^3 + 960x^2y^4 - 384xy^5 + 64y^6\).

Step 1: Determine the Variables in the Binomial Coefficient Formula

In the given binomial expression, \( (2x - 2y)^6 \), \( a = 2x \), \( b = -2y \), and \( n = 6 \).

Step 2: Calculate Binomial Coefficients

For each term in the expansion, calculate the binomial coefficient \( \binom{n}{k} \), where \( n = 6 \) and \( k \) ranges from 0 to 6. The binomial coefficients are as follows: \( \binom{6}{0}, \binom{6}{1}, \binom{6}{2}, \binom{6}{3}, \binom{6}{4}, \binom{6}{5}, \binom{6}{6} \), which evaluate to 1, 6, 15, 20, 15, 6, 1 respectively.

Step 3: Apply the Binomial Theorem

Using the Binomial Theorem, the expanded form of \( (2x - 2y)^6 \) is \(1*(2x)^6*(-2y)^0 + 6*(2x)^5*(-2y) + 15*(2x)^4*(-2y)^2 + 20*(2x)^3*(-2y)^3 + 15*(2x)^2*(-2y)^4 + 6*2x*(-2y)^5 + 1*(-2y)^6\). This simplifies to \(64x^6 - 384x^5y + 960x^4y^2 - 1280x^3y^3 + 960x^2y^4 - 384xy^5 + 64y^6.\)

Key Concepts

Binomial Coefficients

The concept of binomial coefficients is central to understanding how polynomial expressions expand using the Binomial Theorem. Binomial coefficients are used to determine each term's coefficient in the expanded form of a binomial raised to an integer power.

When given an expression like \((a + b)^n\), binomial coefficients are denoted as \(\binom{n}{k}\), which is called a binomial coefficient. Here, \(n\) is the total number of expansions and \(k\) is the specific term’s position, starting from 0.

• Formula for Binomial Coefficient: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
• In the example \((2x - 2y)^6\), the binomial coefficients are: 1, 6, 15, 20, 15, 6, 1, corresponding to terms \(k\) from 0 to 6.

This calculation uses factorials (\(!\)), which are products of all positive integers up to that number, crucial in finding legitimate coefficients in expanded form.

Polynomial Expansion

Polynomial expansion refers to expressing a binomial like \((a + b)^n\) as a sum of terms in the form \(c_k \cdot a^{n-k} \cdot b^k\). Using the binomial theorem, this expansion comprehensively provides all possible products for integers \(n\).

The expansion process involves:

• Identifying the coefficients (using binomial coefficients).
• Determining the powers for \(a\) and \(b\), which decrease and increase respectively from one term to the next.

In our specific problem, \((2x - 2y)^6\), each term is generated by calculating \(\binom{6}{k}\cdot (2x)^{6-k} \cdot (-2y)^k\).

The final result is the polynomial expanded as: \[64x^6 - 384x^5y + 960x^4y^2 - 1280x^3y^3 + 960x^2y^4 - 384xy^5 + 64y^6\].
This shows the complete series once each term's individual component is multiplied out and combined.

Pascal's Triangle

Pascal’s Triangle is a simple yet powerful tool that aids in quickly identifying binomial coefficients for expanding binomials. It mirrors the natural arrangement of coefficients found during polynomial expansion.

The triangle is constructed by stacking rows of numbers, starting with 1 at the top. Each number in the triangle is the sum of the two numbers directly above it from the previous row.

• Top of the triangle starts with a 1.
• Each subsequent row starts and ends with 1.
• Every interior number equals the sum of two numbers diagonally above it.

This assists in finding binomial coefficients without direct calculation of factorials.

For the given problem \((2x - 2y)^6\), using Pascal's Triangle allows students to quickly see the coefficients: 1, 6, 15, 20, 15, 6, 1, without manually computing each with the factorial formula. This visual representation simplifies understanding how the polynomial expansion works naturally.