If f(x3-6x2 11x 8)=X2 1, find the sum of possible values of f(14)?

Question

Answer

Problem Statement

Find the sum of possible values of f(14) if f(x3-6x2+11x+8)=x2+1.

Solution

Let's first simplify the given expression:

f(x3-6x2+11x+8) = x2+1

Let's substitute x = 2:

f(2^3-6(2^2)+11(2)+8) = 2^2+1

f(8-24+22+8) = 5

f(-14) = 5

Let's substitute x = -2:

f((-2)^3-6((-2)^2)+11(-2)+8) = (-2)^2+1

f(-8-24-22+8) = 5

f(-46) = 5

Therefore, the only possible value of f(14) is 5.

Explanation

To solve this problem, we first need to simplify the given expression by substituting x = 2 and x = -2. This gives us two equations, which we can use to find the value of f(-14) and f(-46). Since we know that f(x3-6x2+11x+8) = x2+1, we can substitute x = 14 to find f(14). However, since we only have two equations, we cannot determine the value of f(14) directly. Instead, we can use the fact that f(x) is a polynomial function, which means that it is continuous. Therefore, if f(-14) = 5 and f(-46) = 5, then f(x) must pass through the value of 5 at least once between x = -14 and x = -46. This means that the only possible value of f(14) is 5.

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