Question

Determine if the following equation is an identity.
x⁶+y⁶= (x² + y²) (x⁴ — x²y² +y⁴)

Answer 1

The equation x⁶+y⁶= (x² + y²) (x⁴ − x²y² +y⁴) is an identity because upon expanding the right-hand side, it simplifies to the left-hand side, confirming that the equation holds true for all values of x and y.

Step-by-step explanation:

We are asked to determine if the equation x⁶+y⁶= (x² + y²) (x⁴ − x²y² +y⁴) is an identity. An identity in algebra is an equation that is true for all values of the variables involved.

To verify if the given equation is an identity, we should expand the right-hand side of the equation and see if it equals the left-hand side.

Expanding the right-hand side:

• (x² + y²)(x⁴ - x²y² + y⁴)
• = x²(x⁴ - x²y² + y⁴) + y²(x⁴ - x²y² + y⁴)
• = x⁶ - x⁴y² + x²y⁴ + x⁴y² - x²y⁴ + y⁶
• = x⁶ + y⁶

As we can see, after expanding the right-hand side, we get the left-hand side, which means the equation is indeed an identity.