Polynomials with missing coefficients

Question

Two polynomials are missing coefficients: f(x) = 9x³ + ax² + bx + 1 and g(x) = 9x⁴- cx² - 17x - 2.

The three zeros of f are also zeros of g. Determine f(1)

Answer 1

f(1) = a + b + 10; x₁·x₂·x₃ = - 1/9; - 1/9·x₄ = - 2/9; x₄ = 2; g(2) = 144 - 4c - 34 - 2 = 0; c = 27;

g(x) = 9x⁴ - 27x² - 17x - 2 = 9x⁴ - 36x² + 9x² - 18x + x - 2 = 9x²(x - 2)(x + 2) + 9x(x - 2) + (x - 2) =

(x - 2)(9x³ + 18x² + 9x + 1) = (x - 2)·f(x); f(x) = 9x³ + 18x² + 9x + 1 and f(1) = 37; (a = 18; b = 9)

We apply Viet's theorem