Let f(x) be a quadratic polynomial and g(x) be a linear polynomial, both having integer coefficients. It is given that the coefficient of x^2 in f(g(x)) is 8 , and the coefficient of x in g(f(x)) is 6 . If f(0) = 3 , g(f(0)) = 4 , and f(g(0)) = 5 , then the value of f(2) + g(2) is ______.

Step-by-step solution

Let f(x) = cx^2 + dx + e and g(x) = ax + b , where a, b, c, d, e are integers. Given f(0) = 3 , we have e = 3 . Given g(f(0)) = 4 , we have g(3) = 4 3a + b = 4 . Given f(g(0)) = 5 , we have f(b) = 5 cb^2 + db + 3 = 5 cb^2 + db = 2 . The composition f(g(x)) = c(ax+b)^2 + d(ax+b) + 3 . The coefficient of x^2 in f(g(x)) is ca^2 = 8 . The composition g(f(x)) = a(cx^2+dx+3) + b . The coefficient of x in g(f(x)) is ad = 6 . Since a and c are integers, ca^2 = 8 implies a^2 must divide 8 . The possible integer values for a