If x=2, y=1, and z=−3, then the expression x
3
+y
3
+z
3
−3xyz equals
- A6
- B2
- C8
- D0
Solution & Step-by-step Explanation
Let's look at the algebraic identity for the given expression:
x
3
+y
3
+z
3
−3xyz=(x+y+z)(x
2
+y
2
+z
2
−xy−yz−zx)
First, let's calculate the sum of the variables (x+y+z):
x+y+z=2+1+(−3)
x+y+z=3−3=0
Since the term (x+y+z)=0, substituting this into our identity simplifies the entire product to zero:
x
3
+y
3
+z
3
−3xyz=(0)×(x
2
+y
2
+z
2
−xy−yz−zx)=0
x
3
+y
3
+z
3
−3xyz=(x+y+z)(x
2
+y
2
+z
2
−xy−yz−zx)
First, let's calculate the sum of the variables (x+y+z):
x+y+z=2+1+(−3)
x+y+z=3−3=0
Since the term (x+y+z)=0, substituting this into our identity simplifies the entire product to zero:
x
3
+y
3
+z
3
−3xyz=(0)×(x
2
+y
2
+z
2
−xy−yz−zx)=0