Find a polynomial through given points...?

Question

How do I find an nth order polynomial that goes through specified n points: (x,p(x))?

Answer 1

Let the eqution be

p(x) = a(n-1) x^(n-1)+ a(n-2) x^(n-2) + a

where a(n-1) is subscrpt n-1 for a

put the value of x and P(x) for all the sets of points and you get
n linear equations in a(n-1) to a( 0)

n equations and n unknowns and solve for them

Answer 2

first, you write the polynomial with variables for each term coeficient

example with 3rd order polynomial

p(x)=ax^3 + bx^2 + c^x + d

now you substitue the known points into the equation

for example if one point was (2,3)

3=a(2)^3+b(2)^2+c(2)+d
3=8a+4b+2c+d

if you have 4 points, then you can develop four equations with four unknowns

you can then solve the equation set and get values for a,b,c, and d which can be substituted back into the original equation in x and p(x) to give you the polynomial

you can solve the equation set in several ways including substitution, combination, or matrix algebra (a systematic form of combination that computers and calculaters often use)

Answer 3

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