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

Jasonbradley

2 min read

·

Jun 17, 2024

42

Share

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.