A polynomial f(x)satisfies the equation f(x)+(x+1)^3=2f(x+1) Find f (10)

Answer 1

f(x) must be a monic cubic (why?), so

f(x) = x^3+ax^2+bx+c
f(x+1) = (x+1)^3 + a(x+1)^2 + b(x+1) + c
= x^3 + (3+a)x^2 + (3+2a+b)x + (1+a+b+c)

So,
f(x)+(x+1)^3 = 2x^3 + (3+a)x^2 + (3+b)x + (1+c)

3+a = 2(3+a)
3+b = 2(3+2a+b)
1+c = 2(1+a+b+c)

a = -3
b = 9
c = -13

f(x) = x^3 - 3x^2 + 9x - 13

Check:
f(x+1) = x^3 + 6x^2 - 6
f(x)+(x+1)^3 = 2x^3 + 12x - 12 = 2f(x+1)