Roads can curve too

[BMAD]

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)?

[superprismatic]

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.

[superprismatic]

a) 4

b) y = x⁴ - (89/6)x³ + (157/2)x² - (524/3)x +141
c) No. But it does go through (1,31)

[BMAD]

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?