Question 1209724: Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute p^2 qrs + pq^2 rs + pqr^2 s + pqrs^2.
Found 3 solutions by ikleyn, Edwin McCravy, mccravyedwin:
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website! .
Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute p^2*qrs + pq^2*rs + pqr^2*s + pqrs^2.
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Reduce the polynomial to the standard form, combining like terms
3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14 = x^4 + 2x^3 + 16x^2 + 20x - 31.
Notice that
p^2*qrs + pq^2*rs + pqr^2*s + pqrs^2 = pqrs*(p + q + r + s).
Also notice that the leading coefficient of the polynomial standard form at x^4 is 1.
Due to Vieta's theorem
pqrs = -31 (the product of the roots is equal to the constant term)
p + q + r + s = -2 (the sum of the roots is the coefficient at x^3 with the opposite sign).
Therefore,
p^2*qrs + pq^2*rs + pqr^2*s + pqrs^2 = (-31)*(-2) = 62.
ANSWER. p^2*qrs + pq^2*rs + pqr^2*s + pqrs^2 = 62.
At this point, the problem is solved completely.
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Edwin, I explain it to you, again.
The original problem was in this reduced polynomial form.
They composed this their monstrous version in order for to create their OWN NEW problem for their web-site.
- .-.- .-.- .-.- .-.- .-.- .-.- .-.- .-.- .-.- .-.
Edwin, you ask me HOW do I know.
First, it is obvious even for hedgehog or for rabbit.
Second, I have an ability to see through walls.
Really, I can see what happens on the other side of a wall.
I can do it not because the rays of light go through a wall.
I can see, because I know what happens on the other side of a wall.
(using the method of deduction . . . )
Knowledge is a power.
Answer by Edwin McCravy(20054)  (Show Source):
You can put this solution on YOUR website!
But why do they give problem like this:
g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14
instead of like this:
g(x) = x^4 + 2x^3 + 16x^2 + 20x - 31
?????
Edwin
Answer by mccravyedwin(406)  (Show Source):
You can put this solution on YOUR website!
>>>>>Edwin, I explain it to you, again.
>>>>>The original problem was in this reduced polynomial form.
How do you know anything about "the original problem"?
>>>>>They composed this their monstrous version in order for to create their OWN NEW problem for their web-site.
Why would anybody want to put this on their web-site?
It still makes no sense to me.
Edwin
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