(x+y)^3-(x-y)^3-6y(x^2-y^2)

2 min read Jun 17, 2024

Simplifying the Expression: (x+y)^3 - (x-y)^3 - 6y(x^2-y^2)

This article will explore the simplification of the algebraic expression (x+y)^3 - (x-y)^3 - 6y(x^2-y^2). We will use the concepts of binomial expansion, difference of squares, and combining like terms.

Expanding the Cubes

First, we need to expand the cubes using the binomial theorem:

(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3

Substituting the Expansions

Now, let's substitute these expanded expressions back into our original expression:

(x^3 + 3x^2y + 3xy^2 + y^3) - (x^3 - 3x^2y + 3xy^2 - y^3) - 6y(x^2 - y^2)

Simplifying the Expression

We can now simplify by distributing the negative sign and the 6y:

x^3 + 3x^2y + 3xy^2 + y^3 - x^3 + 3x^2y - 3xy^2 + y^3 - 6yx^2 + 6y^3

Combining Like Terms

Finally, we combine like terms:

(x^3 - x^3) + (3x^2y + 3x^2y - 6yx^2) + (3xy^2 - 3xy^2) + (y^3 + y^3 + 6y^3)

This simplifies to:

6x^2y + 8y^3

Conclusion

Therefore, the simplified form of the expression (x+y)^3 - (x-y)^3 - 6y(x^2-y^2) is 6x^2y + 8y^3.