Answer:
Step-by-step explanation:
Let's simplify the given expression \( (x-y)(x+y) + (y+z)(y-z) + (z-x)^2 \):
1. Expand \((x-y)(x+y)\):
\[ (x-y)(x+y) = x^2 - xy + xy - y^2 = x^2 - y^2 \]
2. Expand \((y+z)(y-z)\):
\[ (y+z)(y-z) = y^2 - z^2 \]
3. Expand \((z-x)^2\):
\[ (z-x)^2 = z^2 - 2zx + x^2 \]
Now, substitute these results back into the original expression:
\[ (x-y)(x+y) + (y+z)(y-z) + (z-x)^2 = x^2 - y^2 + y^2 - z^2 + z^2 - 2zx + x^2 \]
Combine like terms:
\[ x^2 - y^2 + y^2 - z^2 + z^2 - 2zx + x^2 = x^2 + x^2 - y^2 + y^2 - z^2 + z^2 - 2zx \]
Simplify further:
\[ 2x^2 - 2zx - y^2 + z^2 \]
So, \( (x-y)(x+y) + (y+z)(y-z) + (z-x)^2 \) simplifies to \( 2x^2 - 2zx - y^2 + z^2 \).
