Question 661685: the equation x^2-x+1=0 has zeros p and q, and equation 3x^2-2x+3= 0 has zeros r and s
without solving the equations compute (p-r)(q-r)(p-s)(q-s)
Answer by kevwill(135)   (Show Source):
You can put this solution on YOUR website! There might be a shorter way to do this, but here's how I solved it.
Given that  has roots at x=p and x=q, we know that
can be factored into  , so equivalently
 .
This gives us  and 
Similarly,  can be factored into  .
We can divide both sides of each by 3 to get
 .
This gives us  and 
The next thing we need is:
And we know from above that  and  so
 or
Similarly, we can get
Finally, we need one last step:
But since and  we have
Now we can multiply out (p-r)(q-r)(p-s)(q-s)
Let's regroup:
Substituting pq=1 and rs=1:
Again, substituting pq=1 and rs=1:
From above,  ,  , and 
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