"How to find the polynomial function whose graph..." solved and explained

High School
Algebra
Pre-Algebra
Equations, expressions, and inequalitie

doctorfinger1ci

Answered question

2022-12-27

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

Answer & Explanation

milanduy19

Beginner2022-12-28Added 8 answers

Simplest solution:

f (x) = 2 x

General solution:

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

Given solution: 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) = 2 x

But keep in mind that there are other polynomial functions that pass 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} - 10 x^{2} + 31 x - 30

Therefore the most general solution is:

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

for any polynomial

P (x)

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