Let x,y,z and be nonzero real numbers. Find all possible values of
\[x|x|+y|y|+z|z|+xy|xy|+xz|xz|+yz|yz|+xyz|xyz|.\]
Thank you
We could use casework for this problem.
First, if x, y, and z are all postive, we have
x|x|+y|y|+z|z|+xyz|xyz|=xx+yy+zz+xyzxyz=1+1+1+1=4
Now, if we have only x is negative, is 0, we have
x|x|+y|y|+z|z|+xyz|xyz|=−|x||x|+yy+zz+−|x|yz|xyz|=−1+1+1−1=0
In the same way, having only y and z being negative will still be 0.
If only z is postive, we have
x|x|+y|y|+z|z|+xyz|xyz|=−|x||x|+−|y||y|+zz+(−|x|)(−|y|)z|xyz|=−1−1+1+1=0
In the same way, if only x and y are positive, the result will also be 0.
If all three x, y, and z are negative, we have
x|x|+y|y|+z|z|+xyz|xyz|=−|x||x|+−|y||y|+−|z||z|+(−|x|)(−|y|)(−|z|)|xyz|=−1−1−1−1=−4
So, the three possible values are 0, -4, and 4.
Thanks! :)