Im studying for my test tomorrow on Polynomials, and the practice sheet gives me the polynomial with the points (1,0), (2,10), (3, 42), (4,108) (5, 220), (6, 390).

What is the equation? What is the equation in factored form? And how did you get it?

Relevance

With six points the highest possible degree polynomial that can be determined is a 5th degree. The model for that is:

Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F = y

Take each of the six points, one at a time, and generate an equation by substituting x and y into the model equation:

A + B + C + D + E + F = 0

32A + 16B + 8C + 4D + 2E + F = 10

243A + 81B + 27C + 9D + 3E + F = 42

1024A + 256B + 64C + 16B + 4A + F = 108

3125A + 625B + 125C + 25B + 5A + F = 220

7776A + 1296B + 216C + 36B + 6A + F = 390

Solve these six equations simultaneously:

A = 0

B = 0

C = 2

D = -1

E = -1

F = 0

So the desired polynomial is:

2x^3 - x^2 - x = y

In retrospect, it would have saved effort to have prepared a table of successive differences to determine at the beginning that only a cubic equation was necessary.