Polynomial Complex Worksheets Grade 12

1. Mason is designing a high-frequency antenna system where the impedance Z(s) in ohms is modeled by the cubic polynomial Z(s) = s³ - 7s² + 17s - 15, where s is a complex frequency parameter. To optimize signal transmission, Mason must find all complex values of s for which the impedance is zero. Determine all roots of Z(s) = 0.
2. Isabella is a mechanical engineer analyzing the vibrational modes of a new turbine blade. The displacement of the blade over time is modeled by the polynomial function d(t) = t^3 - 8t^2 + 22t - 20, where t represents time in milliseconds and d(t) represents displacement in millimeters. The blade experiences resonance when the displacement equals zero. Determine all time values when resonance occurs, including any complex solutions that might represent theoretical mathematical behavior in the model.
3. An electrical engineer is analyzing the voltage response in a circuit modeled by the polynomial V(t) = t³ - 4t² + 5t - 2, where t represents time in seconds. The circuit reaches equilibrium when the voltage equals zero. Determine all time values when the voltage is zero, including any complex solutions that represent the complete mathematical behavior of the system.

…and 4 more problems

1. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream using the function C(t) = (t³ - 6t² + 11t - 6)/(t² - 3t + 2), where t represents hours after administration. The drug becomes ineffective when its concentration drops below detectable levels. At what time does the concentration approach zero as t increases indefinitely?
2. A civil engineer is designing a suspension bridge where the vertical displacement of the main cable follows the polynomial function d(x) = x^4 - 8x^3 + 26x^2 - 40x + 25, where x represents the horizontal distance from the left tower in meters and d(x) represents vertical displacement in meters. The engineer needs to find all points where the cable touches the horizontal reference line (d(x) = 0) to ensure proper clearance. Determine all solutions to d(x) = 0, including any complex solutions that might represent mathematical properties of the design.
3. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream using the function C(t) = (t³ - 6t² + 11t - 6)/(t² - 4t + 3), where t represents hours after administration. The engineer needs to determine all time values when the drug concentration becomes undefined due to vertical asymptotes in the model. At what times does this occur?

…and 3 more problems

1. Emma is studying a cubic polynomial f(z) = z³ - 9z² + 33z - 65 and its graph on the complex plane. The graph crosses the real axis at exactly one point, and the other two roots are complex conjugates. Visualize the complex roots as vectors from the origin: one vector ends at point (3, 4i) in the complex plane. What is the magnitude of the vector representing the other complex root?
2. Tane is an aerospace engineer analyzing the stability of a satellite's orbit. The satellite's altitude deviation from its ideal path, in kilometers, is modeled by the polynomial function A(t) = t³ - 9t² + 33t - 45, where t represents time in hours after launch. The satellite is considered stable when its altitude deviation is zero. Determine all time values (including any complex solutions) when the satellite is mathematically at zero deviation, representing either actual stability points or theoretical conditions in the model.
3. Find all complex solutions of x³ - 4x² + 9x - 10 = 0 given that 2 + i is a root.

…and 4 more problems