To solve the given expression [tex]\( z^3 - 5xy + 3xyz \)[/tex] for [tex]\( x = 1 \)[/tex], [tex]\( y = -1 \)[/tex], and [tex]\( z = -1 \)[/tex], let's break it down step-by-step.
1. Calculate [tex]\( z^3 \)[/tex]:
[tex]\[
z = -1 \implies z^3 = (-1)^3 = -1
\][/tex]
Thus, [tex]\( z^3 = -1 \)[/tex].
2. Calculate [tex]\( -5xy \)[/tex]:
[tex]\[
x = 1, y = -1 \implies -5xy = -5 \cdot 1 \cdot -1 = 5
\][/tex]
So, [tex]\( -5xy = 5 \)[/tex].
3. Calculate [tex]\( 3xyz \)[/tex]:
[tex]\[
x = 1, y = -1, z = -1 \implies 3xyz = 3 \cdot 1 \cdot -1 \cdot -1 = 3
\][/tex]
Therefore, [tex]\( 3xyz = 3 \)[/tex].
4. Combine the results:
[tex]\[
z^3 - 5xy + 3xyz = -1 + 5 + 3
\][/tex]
5. Sum up the terms:
[tex]\[
-1 + 5 + 3 = 7
\][/tex]
So, the value of the expression [tex]\( z^3 - 5xy + 3xyz \)[/tex] when [tex]\( x = 1 \)[/tex], [tex]\( y = -1 \)[/tex], and [tex]\( z = -1 \)[/tex] is [tex]\( \boxed{7} \)[/tex].
