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 .\
[THOUGHT] We need the coefficient of x^3 in h(x)=f(x+1)-g(x+2). The condition is f(x)≠g(x) for all real x, so f-g has no real zeros. Since f and g share x^4 term, f(x)-g(x) is at most cubic; but any real cubic must have a real root unless its leading coefficient is 0. Hence a3=b3. Then compute the x^3 coefficient in h(x) and square it. [/THOUGHT] [PYTHON] import sympy as sp
x=sp.Symbol('x', real=True) a1,a2,a3,b1,b2,b3=sp.symbols('a1 a2 a3 b1 b2 b3', real=True)
f = a1 + 10x + a2x2 + a3*x3 + x4 g = b1 + 3x + b2x2 + b3*x3 + x4 h = sp.expand(f.subs(x, x+1) - g.subs(x, x+2))
coeff_x3 = sp.expand(h).coeff(x,3) print("h(x) =", h) print("coeff of x^3 =", coeff_x3)
apply necessary condition for f(x) != g(x) for all real x
coeff_x3_conditioned = sp.simplify(coeff_x3.subs({a3:b3})) print("coeff of x^3 under a3=b3 =", coeff_x3_conditioned) print("square =", sp.expand(coeff_x3_conditioned**2)) [/PYTHON]
[THOUGHT] From the verified expansion, the coefficient of (x^3) in (h(x)) is (a_3-b_3-4). The condition (f(x)\ne g(x)) for all real (x) forces (a_3=b_3), so the coefficient is (-4), whose square is (16). [\boxed{16}]