(3x-5y)^3-(5x-2y)^3+(2x+3y)^3

4 min read Jun 16, 2024

Factoring and Simplifying the Expression (3x-5y)^3-(5x-2y)^3+(2x+3y)^3

This article will explore the factorization and simplification of the expression: (3x-5y)^3-(5x-2y)^3+(2x+3y)^3.

Understanding the Problem

The expression involves the difference of cubes pattern. This pattern is a helpful tool for factoring expressions of the form a^3 - b^3. It states:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

We can apply this pattern to simplify our expression.

Applying the Difference of Cubes Pattern

Let's break down the expression and apply the difference of cubes pattern:

(3x-5y)^3-(5x-2y)^3:
- Here, a = 3x-5y and b = 5x-2y
- Applying the pattern, we get:
  - [(3x-5y) - (5x-2y)][(3x-5y)^2 + (3x-5y)(5x-2y) + (5x-2y)^2]
(2x+3y)^3:
- We can't directly apply the difference of cubes pattern to this term since it is not in the form a^3 - b^3.

Simplifying the Expression

Now, let's simplify the factored expression:

Simplify the first factor:
- (3x-5y) - (5x-2y) = -2x - 3y
Expand the second factor:
- (3x-5y)^2 = 9x^2 - 30xy + 25y^2
- (3x-5y)(5x-2y) = 15x^2 - 29xy + 10y^2
- (5x-2y)^2 = 25x^2 - 20xy + 4y^2
Combine the terms in the second factor:
- 9x^2 - 30xy + 25y^2 + 15x^2 - 29xy + 10y^2 + 25x^2 - 20xy + 4y^2 = 49x^2 - 79xy + 39y^2
Combine the first and second factors:
- (-2x - 3y)(49x^2 - 79xy + 39y^2) + (2x+3y)^3
Expand the final term:
- (2x+3y)^3 = 8x^3 + 36x^2y + 54xy^2 + 27y^3
Combine all terms:
- (-2x - 3y)(49x^2 - 79xy + 39y^2) + 8x^3 + 36x^2y + 54xy^2 + 27y^3

Conclusion

The fully simplified expression is: (-2x - 3y)(49x^2 - 79xy + 39y^2) + 8x^3 + 36x^2y + 54xy^2 + 27y^3. While this expression appears complex, it is important to note that it has been significantly simplified by using the difference of cubes pattern. This expression can be further simplified by multiplying out the terms, but it will likely result in a longer and more complicated expression.