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Roads can curve too

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A couple weeks ago, I created a question requiring the shortest path. Now for this question, assume that roads can be curved. We need a road that can pass through the following four cities (location of each city listed as coordinates): Los Angeles (3,4), Newport Beach (5,1), Pasadena (4,5), Santa Monica (2,3).

a) What's the smallest degree polynomial y=f(x) that will pass through all four cities? . . .
b) What is the exact equation of this polynomial? (Hint: use fractions not decimals)
c) Would this road go through Chatsworth at (1,6)?

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3 answers to this question

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a) 3


b) y = (-5/6)x3 + (15/2)x2 - (62/3)x + 21
c) No. But it does go through (1,7)

I tried solving for the coefficients of a quadratic using gaussian elimination
without success, as the system of linear equations was inconsistent. When I
did the same thing with the coefficients of a cubic, I found the system to be
consistent with the coefficients (-5/6), (15/2), (-62/3), and 21. Previously,

I had erred in solving the cubic case and went on to the quartic with success.

The quartic I had in post #2 is the smallest degree monic polynomial that does

the job. Monic, however, was not a requirement.

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a) 4

b) y = x4 - (89/6)x3 + (157/2)x2 - (524/3)x +141

c) No. But it does go through (1,31)

care to explain your approach?

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