A polynomial f(x)satisfies the equation f(x)+(x+1)^3=2f(x+1) Find f (10)
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)
To find the value of f(10), we need to solve the given equation step by step. Let's start by simplifying the equation:
f(x) + (x + 1)^3 = 2f(x + 1)
Expanding (x + 1)^3 on the left side of the equation, we have:
f(x) + (x^3 + 3x^2 + 3x + 1) = 2f(x + 1)
Next, let's try to simplify the equation further by collecting like terms. Since we want to find the value of f(10), let's substitute x = 9 into the equation:
f(9) + (9^3 + 3(9)^2 + 3(9) + 1) = 2f(10)
Now, we can see that the equation involves the unknown value f(10) as well as f(9). To solve for f(10), we need to know the value of f(9). Unfortunately, the equation itself does not provide enough information to directly calculate f(9) or f(10).
Therefore, without any additional information about the polynomial function or its values at specific points, we cannot determine the exact value of f(10).