(xy+1)(x^2y^2-xy+1)+(x^3-1)(1-y^3)=x^3+y^3

2 min read Jun 17, 2024
(xy+1)(x^2y^2-xy+1)+(x^3-1)(1-y^3)=x^3+y^3

Expanding and Simplifying the Equation: (xy+1)(x^2y^2-xy+1)+(x^3-1)(1-y^3)=x^3+y^3

This equation appears complex, but it can be simplified through algebraic manipulation. Let's break down the process step by step:

1. Expanding the Products

First, we need to expand the products on the left side of the equation using the distributive property (also known as FOIL method):

  • (xy + 1)(x²y² - xy + 1):

    • xy * x²y² = x³y³
    • xy * (-xy) = -x²y²
    • xy * 1 = xy
    • 1 * x²y² = x²y²
    • 1 * (-xy) = -xy
    • 1 * 1 = 1
  • (x³ - 1)(1 - y³):

    • x³ * 1 = x³
    • x³ * (-y³) = -x³y³
    • -1 * 1 = -1
    • -1 * (-y³) = y³

2. Combining Like Terms

Now, let's combine the terms we obtained from expanding the products:

x³y³ - x²y² + xy + x²y² - xy + 1 + x³ - x³y³ - 1 + y³ = x³ + y³

Notice that several terms cancel out:

  • x³y³ and -x³y³ cancel
  • -x²y² and x²y² cancel
  • xy and -xy cancel
  • 1 and -1 cancel

This leaves us with:

x³ + y³ = x³ + y³

3. Conclusion

The simplified equation is x³ + y³ = x³ + y³. This is a true statement for any values of x and y. Therefore, the original equation is an identity, meaning it holds true for all values of its variables.

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