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You have four points, so for them to fit the polynomial exactly, you need it to at least have degree three. Anything more you'll have an infinite number of possibilities that will have all data points fit, and anything less then chances are you'll only be able to get a least squares approximation.If p(x) is a polynomial such that p(0)=5 ,p(1)=4 ,p(2)=9,p(3)=20 ,
the minimum degree it can have
If [tex]p(x)[/tex] is a polynomial such that: .[tex]p(0) = 5,\;p(1) = 4,\;p(2) = 9,\;p(3) = 20,[/tex]
. . the minimum degree it can have is __.
Then polynomial, passing through four given points, will have degree at most three, not "at least". It is quite possible that the four points happen to lie on a parabola (which is apparently the case here) or even on a straight line.You have four points, so for them to fit the polynomial exactly, you need it to at least have degree three. Anything more you'll have an infinite number of possibilities that will have all data points fit, and anything less then chances are you'll only be able to get a least squares approximation.
Really? I would have thought that there would be an infinite number of solutions to, say, four equations in five unknowns, which is what you would get if you substituted the four points into a general polynomial of degree 4...Then polynomial, passing through four given points, will have degree at most three, not "at least". It is quite possible that the four points happen to lie on a parabola (which is apparently the case here) or even on a straight line.
Four equations in five unknowns is what you will end up with when you try to fit a quartic, and we know you can always do that but the solution is not unique.Really? I would have thought that there would be an infinite number of solutions to, say, four equations in five unknowns, which is what you would get if you substituted the four points into a general polynomial of degree 4...