Question 1209721: For parts (a)-(d), 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 pqr + pqs + prs + qrs.
Answer by math_tutor2020(3816)   (Show Source):
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Answer: -20
Explanation
3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14
simplifies to
x^4 + 2x^3 + 16x^2 + 20x - 31
Consider the general quartic function
f(x) = ax^4 + bx^3 + cx^2 + dx + e
According to Vieta's formulas (specifically formulas (4) and (5) on that page) we can state that,
pqr + pqs + prs + qrs = -d/a = -20/1 = -20
Another approach is to determine the approximate values of p,q,r,s
p = -2.1259
q = 0.8627
r = -0.3684 + 4.0948i
s = -0.3684 - 4.0948i
The order of the roots doesn't matter.
Then use a calculator to get this approximate result
pqr + pqs + prs + qrs = -20.0007015303
If you were to use more decimal digits in each of the four roots, then you would get closer to -20
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