taiwan-gsat 2024 Q5

taiwan-gsat · Other · ast__math-a 8 marks Complex Numbers Arithmetic True/False or Property Verification Statements
Let $f(x)$ be a cubic polynomial with real coefficients. It is known that $f(-2 - 3i) = 0$ (where $i = \sqrt{-1}$), and the remainder when $f(x)$ is divided by $x^{2} + x - 2$ is 18. Select the correct options.
(1) $f(2 + 3i) = 0$
(2) $f(-2) = 18$
(3) The coefficient of the cubic term of $f(x)$ is negative
(4) $f(x) = 0$ has exactly one positive real root
(5) The center of symmetry of the graph $y = f(x)$ is in the first quadrant
Let $f(x)$ be a cubic polynomial with real coefficients. It is known that $f(-2 - 3i) = 0$ (where $i = \sqrt{-1}$), and the remainder when $f(x)$ is divided by $x^{2} + x - 2$ is 18. Select the correct options.\\
(1) $f(2 + 3i) = 0$\\
(2) $f(-2) = 18$\\
(3) The coefficient of the cubic term of $f(x)$ is negative\\
(4) $f(x) = 0$ has exactly one positive real root\\
(5) The center of symmetry of the graph $y = f(x)$ is in the first quadrant