How do you use pascal’s triangle to expand (2x - 3y)³?

To expand the expression (2x - 3y)³ using Pascal's Triangle, follow these steps:

Understanding Pascal's Triangle

Pascal's Triangle is a triangular array of numbers that helps in binomial expansions. Each row corresponds to the coefficients of the binomial expansion for increasing powers.

Identifying the Coefficients

For (a + b)³, the coefficients from Pascal's Triangle are found in the fourth row: 1, 3, 3, 1. These coefficients will be used for the expansion.

Applying the Binomial Theorem

The binomial theorem states that:

• (a + b)ⁿ = Σ [C(n, k) * a^(n-k) * b^k]

Where C(n, k) is the binomial coefficient.

Expanding (2x - 3y)³

In this case, let a = 2x and b = -3y. Now, we can apply the coefficients:

• 1 * (2x)³ * (-3y)⁰
• + 3 * (2x)² * (-3y)¹
• + 3 * (2x)¹ * (-3y)²
• + 1 * (2x)⁰ * (-3y)³

Calculating Each Term

Now, calculate each term:

• 1 * 8x³ * 1 = 8x³
• 3 * 4x² * (-3y) = -36x²y
• 3 * 2x * 9y² = 54xy²
• 1 * 1 * (-27y³) = -27y³

Final Result

Combining all the terms gives:

8x³ - 36x²y + 54xy² - 27y³

This is the expanded form of (2x - 3y)³ using Pascal's Triangle.