Let denote the set of all real numbers. Let for .Define the functions , , and byf(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4,$$$$g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4,$$$$h(x) = f(x+1) - g(x+2).If for every , find the square of the coefficient of in .\
The coefficient of (x^3) in (h(x)) is (a_3 - b_3 - 4). When (a_i = 0) and (b_i = 0) for all (i), (f(x) = 10x + x^4) and (g(x) = 3x + x^4), and (f(x) \neq g(x)) for every (x) since the linear coefficients differ. For this case, (f(x+1) = (x+1)^4 + 10(x+1) = x^4 + 4x^3 + 6x^2 + 14x + 11) and (g(x+2) = (x+2)^4 + 3(x+2) = x^4 + 8x^3 + 24x^2 + 35x + 22), so (h(x) = f(x+1) - g(x+2) = -4x^3 - 18x^2 - 21x - 11). The coefficient of (x^3) is (-4), and its square is (16).
\boxed{16}
[THOUGHT] I need to compute the coefficient of (x^3) in (h(x) = f(x+1) - g(x+2)) for the given functions, using Python. I'll define the functions with specific values for (a_i) and (b_i) that satisfy the condition (f(x) \neq g(x)) for all (x), then expand and extract the coefficient. [/THOUGHT] [PYTHON] from sympy import symbols, expand
x = symbols('x') a1, a2, a3 = 0, 0, 0 b1, b2, b3 = 0, 0, 0
f = a1 + 10x + a2x2 + a3*x3 + x4 g = b1 + 3x + b2x2 + b3*x3 + x4
h = f.subs(x, x+1) - g.subs(x, x+2) h_expanded = expand(h) coeff = h_expanded.coeff(x, 3) print(coeff) print(coeff**2) [/PYTHON]
\boxed{16}