(2x-3y)^4

4 min read Jun 16, 2024

Expanding (2x-3y)^4: A Step-by-Step Guide

Expanding expressions like (2x-3y)^4 can seem daunting, but with the right method, it becomes manageable. This article provides a clear breakdown of the process, using the Binomial Theorem and exploring the Pascal's Triangle as a helpful tool.

The Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions of the form (a + b)^n. It states:

(a + b)^n = ∑(n choose k) a^(n-k) b^k

where:

n is the power to which the binomial is raised.
k is an integer ranging from 0 to n.
(n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

Applying the Binomial Theorem to (2x-3y)^4

Let's apply the Binomial Theorem to our expression (2x-3y)^4:

Identify a and b: In this case, a = 2x and b = -3y.
Set n = 4: We are raising the binomial to the power of 4.
Expand using the Binomial Theorem:

(2x - 3y)^4 = ∑(4 choose k) (2x)^(4-k) (-3y)^k
Calculate the binomial coefficients:
- (4 choose 0) = 1
- (4 choose 1) = 4
- (4 choose 2) = 6
- (4 choose 3) = 4
- (4 choose 4) = 1
Substitute the values and simplify:

(2x - 3y)^4 = (1)(2x)^4 (-3y)^0 + (4)(2x)^3 (-3y)^1 + (6)(2x)^2 (-3y)^2 + (4)(2x)^1 (-3y)^3 + (1)(2x)^0 (-3y)^4

= 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4

Pascal's Triangle: A Visual Aid

Pascal's Triangle provides a convenient way to visualize the binomial coefficients. Each row corresponds to a power of the binomial, and the numbers represent the binomial coefficients.

For (2x-3y)^4, we would look at the 5th row of Pascal's Triangle:

        1
      1   1
    1   2   1
  1   3   3   1
1   4   6   4   1

The numbers in the 5th row (1 4 6 4 1) correspond to the binomial coefficients we calculated previously.

Conclusion

Expanding (2x-3y)^4 using the Binomial Theorem is a systematic process. Understanding the theorem and utilizing Pascal's Triangle for visual aid can greatly simplify the expansion. This process can be generalized to any binomial raised to any power.