Question

Given that f (0) = 1, f (1) = 3, f (3) = 55, find the unique polynomial of degree 2 or less, which fits the given data.

Answer 1

The standard form of 2 degree polynomial is ax^2+bx+c

then,

f(x) = ax^2+bx+c

f(0) = 0 +0 + c

f(0) = c

so there is given that f(0) = 1

so c= 1

then,

f(x)= ax^2+bx+c

f(1)= a+b+c

there is given that f(1)= 3 , so

a+b+c= 3

a+b+1= 3 ( putting the value of c)

a+b= 2so then

f(x)= ax^2+bx+c

f(3)= 9a+3b+c

there is given f(3) = 55

then,

9a+3b+c=55

9a+3b+1= 55

9a+3b= 54

3(3a+b) = 54

3a+b = 54/3

2a+a+b= 18

2a+2= 18 ( putting the value of a+b)

2a=16

a=8 , so only bvalue left

a+b=2

8+b=2

b= -6

so it fit for that

so the polynomial is ax^2+bx+c

8x^2-6x+c