SOLUTION: 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)

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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