for the expression f (x) = x^3 + ax^2 + bx + c , if f(1) = f(2) = 0 and f(4) = f(0) . then a+b+ c = 

Given: f (x) = x³ + ax² + bx + c, f(1) = f(2) = 0 and f(4) = f(0)

Now, f(1) = f(2) = 0

i.e. a + b + c = -1   ... (i)

Also, 4a + 2b + c = -8   ... (ii)

As f(4) = f(0)

64 + 16a + 4b + c = c

i.e. 4a + b = -16   ... (iii)

Putting this in (ii), we get

-16 + b + c = -8

i.e. b + c = 8

Putting this value in (i), we get

a + 8 = -1

i.e. a = -9

Therefore, b = -16 + 36 = 20

Thus, c = -12

Hence, a + b + c = -9 + 20 - 12 = -1