Polynomials

Question

Determine the 3rd degree polynomial function that passes through the following four points:


(1, 2), (2, 15), (-1, -12), and (0, 3).


What is the value of a2 (the coefficient of x2)?

Answer 1

Note that in general two points determine a line (that is, given two points, you can write the exact equation for the line containing the two points).

 

If we step up a degree to parabola (degree 2), in general, we need 3 points to determine a parabola.

 

In general, N+1 points are sufficient to determine an Nth degree polynomial.

 

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Now, for the problem you mention, we know the form of the function whose equation we seek (cubic): y=A₃x³+A₂x²+A₁x+A₀ where the "A_k" terms are coefficients of the k-th degree term.

 

Moreover, using this as a template, we can write equations for each coordinate provided to mathematically represent the fact that these points lie on the function; thus, we will have 4 equations.

 

We also have 4 unknowns (all the coefficients are unknown).

 

A linear system of 4 variables in 4 equations is what we have.