Let f(x) be a quadratic polynomial with real coefficients such that f(f(x)) = 8x^4 - 8x^3 + 8x^2 - 3x + 2 . Then the value of f(3) is ______.

Step-by-step solution

Let f(x) = ax^2 + bx + c . Then f(f(x)) = a(ax^2+bx+c)^2 + b(ax^2+bx+c) + c . Expanding the terms, we get: f(f(x)) = a(a^2x^4 + 2abx^3 + (b^2+2ac)x^2 + 2bcx + c^2) + abx^2 + b^2x + bc + c = a^3x^4 + 2a^2bx^3 + (ab^2 + 2a^2c + ab)x^2 + (2abc + b^2)x + (ac^2 + bc + c) . Given f(f(x)) = 8x^4 - 8x^3 + 8x^2 - 3x + 2 . Comparing the coefficients of x^4 : a^3 = 8 a = 2 . Comparing the coefficients of x^3 : 2a^2b = -8 2(4)b = -8 8b = -8 b = -1 . Comparing the coefficients of x^2 : ab^2 + 2a^2c + ab = 8 Substituting a = 2 a