To solve the expression [tex]\((2x - 5y)^3 - (2x + 5y)^3\)[/tex], let's follow these steps.
1. Understand the expression: We are dealing with a subtraction of two cubes:
[tex]\[
(2x - 5y)^3 - (2x + 5y)^3
\][/tex]
2. Use the algebraic identity for the difference of cubes: The difference of two cubes [tex]\(a^3 - b^3\)[/tex] can be factored using the identity:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
Here, [tex]\(a = 2x - 5y\)[/tex] and [tex]\(b = 2x + 5y\)[/tex].
3. Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the identity:
[tex]\[
(2x - 5y)^3 - (2x + 5y)^3 = [(2x - 5y) - (2x + 5y)][(2x - 5y)^2 + (2x - 5y)(2x + 5y) + (2x + 5y)^2]
\][/tex]
4. Simplify the first binomial:
[tex]\[
(2x - 5y) - (2x + 5y) = 2x - 5y - 2x - 5y = -10y
\][/tex]
5. Simplify the second part:
[tex]\[
(2x - 5y)^2 = 4x^2 - 20xy + 25y^2
\][/tex]
[tex]\[
(2x - 5y)(2x + 5y) = 4x^2 - 25y^2
\][/tex]
[tex]\[
(2x + 5y)^2 = 4x^2 + 20xy + 25y^2
\][/tex]
6. Combine these results:
[tex]\[
(2x - 5y)^2 + (2x - 5y)(2x + 5y) + (2x + 5y)^2 = 4x^2 - 20xy + 25y^2 + 4x^2 - 25y^2 + 4x^2 + 20xy + 25y^2
\][/tex]
7. Simplify the combination of terms:
[tex]\[
4x^2 - 20xy + 25y^2 + 4x^2 - 25y^2 + 4x^2 + 20xy + 25y^2 = 4x^2 + 4x^2 + 4x^2 + 25y^2 - 25y^2 + 25y^2 = 12x^2 + 25y^2
\][/tex]
8. Multiply the simplified factors:
[tex]\[
-10y \cdot (12x^2 + 25y^2) = -10y \cdot 12x^2 - 10y \cdot 25y^2 = -120x^2y - 250y^3
\][/tex]
Hence, the final result of the given expression [tex]\((2x - 5y)^3 - (2x + 5y)^3\)[/tex] is:
[tex]\[
-120x^2y - 250y^3
\][/tex]
