Find the unique n 1th degree polynomials that pass through

Find the unique (n - 1)-th degree polynomials that pass through the given sets of n data points. (a) The data points: (0,2), (1,4), (2,6) (b) The data points: (-1,0), (0,3), (1,6), (2,15)

Solution

a) we have 3 points ( 0, 2) , (1, 4) and (2.6)

we can find a quadratic polynomial :

y = ax^2 +bx +c

plug the points to form equations:

2 = 0 +0 +c ----> c =2

4= a +b +2 ----> a +b = 2

6 = 4a +2b +2 ----> 4a +2b = 4

a +b =2

2a +b = 2

-------------------- (subtract the two)

-a =0 ; b =2

So, y = 2x +2 ( polynomial)

b) ( we have 4 points we would find cubic polynomail

y = ax^3 +bx^2 +cx +d

(0, 3) ; 3 = 0 +0 +0 +d ----> d =3

( -1 , 0) ; 0 = -a +b -c +3 ----> -a +b -c = -3 -----(1)

(1, 6) ; 6 = a +b +c +3 ----> a +b +c = 3 ----(2)

(2, 15) ; 15 = 8a +4b +2c +3 ---> 8a +4b +2c = 12 ---(3)

solve the three eqautions: we get a = 3 ; b =-6 ; c =6

So, y = 3x^3 -6x^2 +6x +3   ( polynomial)