How do you find the polynomial function whose graph passes through (2,4), (3,6), (5,10)?

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Answer 1 · 2018-05-19T07:11:38

Simplest solution:

#f(x) = 2x#

General solution:

#f(x) = P(x)(x^3-10x^2+31x-30)+2x#

Given:

#(2, 4)#, #(3, 6)#, #(5, 10)#

Note that each #y# coordinate is twice the corresponding #x# coordinate.

So a suitable polynomial function is:

#f(x) = 2x#

Note however that this is not the only polynomial function passing through these three points.

We can add any multiple (scalar or polynomial) of a cubic whose zeros lie at those three points, namely:

#(x-2)(x-3)(x-5) = x^3-10x^2+31x-30#

Hence the most general solution is:

#f(x) = P(x)(x^3-10x^2+31x-30)+2x#

for any polynomial #P(x)#.