Here is a step-by-step explanation of simplifying the expression [tex]\( x y z \left(x^2 - y^2 - z^2\right) \)[/tex]:
1. Identify the Expression: The given expression is [tex]\( x y z \left(x^2 - y^2 - z^2\right) \)[/tex].
2. Expand the Product:
To simplify the expression, we first expand the product inside the parentheses:
[tex]\[
x^2 - y^2 - z^2
\][/tex]
3. Distribute [tex]\( x y z \)[/tex]:
Next, distribute [tex]\( x y z \)[/tex] to each term inside the parentheses:
[tex]\[
x y z (x^2 - y^2 - z^2) = x y z \cdot x^2 - x y z \cdot y^2 - x y z \cdot z^2
\][/tex]
4. Multiply Each Term:
Perform the multiplication for each term:
[tex]\[
x y z \cdot x^2 = x^3 y z
\][/tex]
[tex]\[
x y z \cdot y^2 = x y^3 z
\][/tex]
[tex]\[
x y z \cdot z^2 = x y z^3
\][/tex]
5. Combine the Terms:
Combine the terms to write out the final expanded expression:
[tex]\[
x y z \left(x^2 - y^2 - z^2\right) = x^3 y z - x y^3 z - x y z^3
\][/tex]
Thus, by following these steps, we break down the given expression and simplify it accordingly.
