Solving the Expression ( frac{x^2}{2}yz frac{y^2}{2}xz frac{z^2}{2}xy ) Given ( x y z 0 )
In this tutorial, we will explore the detailed steps to simplify the expression:
Expression and Condition
We are given the expression:
(frac{x^2}{2}yz frac{y^2}{2}xz frac{z^2}{2}xy)
and the condition:
(x y z 0)
Step-by-Step Solution
To begin, let's start by rewriting Z in terms of X and Y.
Step 1: Substitute Z in Terms of X and Y
From the condition x y z 0, we can express z as:
(z -x - y)
Step 2: Substitute Z into the Expression
Now, substitute z -x - y into the original expression:
First Term: (frac{x^2}{2}yz)
(frac{x^2}{2}yz) becomes:
(frac{x^2}{2}y(-x - y) frac{x^2}{2}(-xy - y^2) -frac{x^3y}{2} - frac{x^2y^2}{2})
Second Term: (frac{y^2}{2}xz)
(frac{y^2}{2}xz) becomes:
(frac{y^2}{2}x(-x - y) frac{y^2}{2}(-x^2 - xy) -frac{x^2y^2}{2} - frac{xy^3}{2})
Third Term: (frac{z^2}{2}xy)
(frac{z^2}{2}xy) becomes:
(frac{(-x - y)^2}{2}xy frac{x^2 2xy y^2}{2}xy frac{x^3y 2x^2y^2 y^3x}{2})
Step 3: Combine All Terms
Now, combine all the terms:
(-frac{x^3y}{2} - frac{x^2y^2}{2} - frac{x^2y^2}{2} - frac{xy^3}{2} frac{x^3y}{2} frac{x^2y^2}{2} frac{x^2y^2}{2} frac{xy^3}{2})
Observe that the terms (frac{x^3y}{2}), (-frac{x^3y}{2}), (-frac{x^2y^2}{2}), (-frac{x^2y^2}{2}), (frac{x^2y^2}{2}), (frac{x^2y^2}{2}), (frac{x^3y}{2}), and (-frac{x^3y}{2}) cancel each other out.
Similarly, the terms (-frac{xy^3}{2}), and (frac{xy^3}{2}) also cancel out.
Step 4: Final Simplification
After all cancellations, the expression simplifies to:
(0)
Conclusion
Therefore, the final result is: