Problem Statement
For certain real numbers $a$, $b$, and $c$, the polynomial $g(x) = x^3 + ax^2 + x + 10$ has three distinct roots, and each root of $g(x)$ is also a root of the polynomial $f(x) = x^4 + x^3 + bx^2 + 100x + c$. What is $f(1)$?
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Solution
$$g(x) = x^3 + ax^2 + x + 10 = (x - r_1)(x - r_2)(x - r_3)$$ $$g(x) = x^3 + ax^2 + x + 10 = x^3 - (r_1 + r_2 + r_3)x^2 + (r_1r_2 + r_2r_3 + r_1r_3)x - r_1r_2r_3$$ $$\implies - (r_1 + r_2 + r_3) = a, (r_1r_2 + r_2r_3 + r_1r_3) = 1, - r_1r_2r_3 = 10$$ $$\implies (r_1 + r_2 + r_3) = -a, (r_1r_2 + r_2r_3 + r_1r_3) = 1, r_1r_2r_3 = -10$$ $$f(x) = x^4 + x^3 + bx^2 + 100x + c = (x - r_1)(x - r_2)(x - r_3)(x - r_4)$$ $$f(x) = x^4 + x^3 + bx^2 + 100x + c = x^4 - (r_1 + r_2 + r_3 + r_4)x^3 + (r_1r_2 + r_1r_3 + r_2r_3 + r_4(r_1 + r_2 + r_3))x^2 - (r_1r_2r_3 + r_4(r_1r_2 + r_1r_3 + r_2r_3))x + r_1r_2r_3r_4$$ $$\implies - (r_1 + r_2 + r_3 + r_4) = 1, (r_1r_2 + r_1r_3 + r_2r_3 + r_4(r_1 + r_2 + r_3)) = b, -(r_1r_2r_3 + r_4(r_1r_2 + r_1r_3 + r_2r_3)) = 100, r_1r_2r_3r_4 = c$$ $$-(r_1r_2r_3 + r_4(r_1r_2 + r_1r_3 + r_2r_3)) = 100 \implies -(-10 + r_4) = 100 \implies r_4 = -90$$ $$r_1r_2r_3r_4 = c \implies (-10)(-90) = c = 900$$ $$- (r_1 + r_2 + r_3 + r_4) = 1 \implies -(-a - 90) = 1 \implies a = -89$$ $$(r_1r_2 + r_1r_3 + r_2r_3 + r_4(r_1 + r_2 + r_3)) = b \implies 1 + (-90)(-(-89)) = -8009 = b$$ $$\therefore f(x) = x^4 + x^3 -8009x^2 + 100x + 900$$ $$\therefore f(1) = (1)^4 + (1)^3 - 8009(1^2) + 100(1) + 900 = 1 + 1 - 8009 + 100 + 900 = -7007$$ 