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Intersections of a line and a curve

by ehrenfest
Tags: curve, intersections, line
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ehrenfest
#1
Jul21-07, 09:40 PM
P: 1,996
1. The problem statement, all variables and given/known data

For all all lines which meet the graph y = 2x^4+7x^3+3x-5 at four distinct points, what is the sum of the x-coordinates of the four points of intersection?


2. Relevant equations



3. The attempt at a solution

So, you obviously set ax+b =2x^4+7x^3+3x-5, but how do you find the zero's of that?
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Hurkyl
#2
Jul21-07, 09:47 PM
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You don't need to know the roots; you only need to know their sum.
ehrenfest
#3
Jul22-07, 12:35 AM
P: 1,996
Yes and how do you know their sum without knowing the roots?

Gib Z
#4
Jul22-07, 03:35 AM
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Intersections of a line and a curve

http://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formulas

Its a bit of grinding but in this case its a degree 4 polynomial so we can get it to the form [itex](x-\alpha)(x-\beta)(x-\gamma)(x-\delta)[/itex]. Where the greek letters are the roots. Now expand that, and equate to co efficients and we can get an expression for the sum of the roots knowing only the coefficients.


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