Question

If f(0) =1,f(1) = 5 and f(2) = 11 ,then equation of polynomial of degree two is?

Answer 1

let f(x) = ax² +bx +c = 0

as , polynomial is of degree 2 , hence it is a quadratic equation .

so , given => f (0) = 1

f (1) = 5

f (2) = 11

=> f (0) = a(0)² + b(0) + c

= 0 + 0 + c = 1

=> c = 1 ------------- (i)

=> f (1) = a(1)² + b(1) + c

= a + b + c = 5

=> a + b = 5 - c = 5 - 1 = 4

=> a + b = 4

----------- (ii)

=> f (2) = a(2)² + b (2) + c

=> 11 = 4a + 2b + c

=> 11 - c = 4a + 2b

=> 10 = 2 ( 2a+b ) (as, c=1)

=> 5 = 2a + b --------------- (iii)

solving (ii) and (iii)

=> we get,

2a + b - a - b = 5 - 4

=> a = 1

and putting a = 1 in (ii) ,

=> b = 4 - 1 = 3

therefore , f(x) = ax² + bx + c

=> f (x) = 1(x)² + 3x + 1

=> f (x) = x² + 3x + 1