Help. Use the remainder theorem to find the factors.

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

To find the factors using the remainder theorem, we need to rewrite the expression in a specific form. The remainder theorem states that if we substitute a potential factor into an expression and get a remainder of zero, then that potential factor is indeed a factor of the expression.

Let's start by simplifying the given expression:
(x-y)^2 + (y-z)^3 + (z-x)^3

Expand the squares and cubes:
(x^2 - 2xy + y^2) + (y^3 - 3y^2z + 3yz^2 - z^3) + (z^3 - 3z^2x + 3zx^2 - x^3)

Combining like terms:
x^2 - 2xy + y^2 + y^3 - 3y^2z + 3yz^2 - z^3 + z^3 - 3z^2x + 3zx^2 - x^3

Rearranging the terms:
-x^3 + x^2 + 3zx^2 - 3z^2x - 2xy + y^2 + y^3 - 3y^2z + 3yz^2

Now, we can try potential factors to check if they are indeed factors. In this case, since all the variables are present in each term and they are not common factors, we need to substitute different values for x, y, and z.

Let's start by checking x - y:
Substituting x = y:
-(y)^3 + (y)^2 + 3z(y)^2 - 3z^2(y) - 2y(y) + y^2 + y^3 - 3(y)^2z + 3yz^2

Simplifying further:
- y^3 + y^2 + 3zy^2 - 3z^2y - 2y^2 + y^2 + y^3 - 3y^2z + 3yz^2

As we can see, the terms cancel out and we get a remainder of zero. Therefore, x - y is a factor of the expression.

We can repeat this process by substituting different values for x, y, and z to check potential factors and find other factors of the given expression.