Polynomial Complex

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Polynomial Complex · Grade 12 · Worksheet 2

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 - 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? Answer: t = 1 and t = 3 Solution: Rational functions become undefined when their denominators equal zero, creating vertical asymptotes in their graphs.
   Full step-by-step solution

   Rational functions become undefined when their denominators equal zero, creating vertical asymptotes in their graphs. In mathematical modeling, these points represent scenarios where the model breaks down or becomes physically impossible. To find these critical values, factor both numerator and denominator polynomials completely, then identify the denominator's roots that aren't canceled by identical factors in the numerator. This concept is crucial for understanding the limitations of mathematical models in scientific applications.

2. Sophia is an aerospace engineer analyzing the vibration modes of a new aircraft wing design. The wing's natural frequency response is modeled by the polynomial equation f(x) = x³ - 11x² + 41x - 51, where x represents a dimensionless frequency parameter and f(x) represents the amplitude of vibration. To ensure structural integrity, Sophia must find all frequency parameters where the vibration amplitude is zero, including any complex solutions that represent mathematically possible but physically damped modes. Determine all solutions to f(x) = 0. Answer: x = 3, x = 4 + i, x = 4 - i Solution: Set f(x) = 0: x³ - 11x² + 41x - 51 = 0 Use the Rational Root Theorem. Possible rational roots are factors of 51: ±1, ±3, ±17, ±51. Test x = 3: (3)³ - 11(3)² + 41(3) - 51 = 27 - 99 + 123 - 51 = 0.
   Full step-by-step solution

   Step 1: Set f(x) = 0: x³ - 11x² + 41x - 51 = 0 Step 2: Use the Rational Root Theorem. Possible rational roots are factors of 51: ±1, ±3, ±17, ±51. Step 3: Test x = 3: (3)³ - 11(3)² + 41(3) - 51 = 27 - 99 + 123 - 51 = 0. So x = 3 is a root. Step 4: Use synthetic division with root 3 on coefficients 1, -11, 41, -51: Bring down 1. 1 * 3 = 3, add to -11: -8. -8 * 3 = -24, add to 41: 17. 17 * 3 = 51, add to -51: 0. Step 5: The quotient is x² - 8x + 17. Step 6: Solve x² - 8x + 17 = 0 using the quadratic formula: x = [8 ± sqrt(64 - 68)]/2 = [8 ± sqrt(-4)]/2 = [8 ± 2i]/2 = 4 ± i. Step 7: The complete solution set is x = 3, x = 4 + i, x = 4 - i.

3. An electrical engineer is analyzing the voltage response in a circuit using the polynomial function V(t) = t³ - 4t² + 6t - 4, where t represents time in seconds and V(t) represents voltage in volts. The circuit reaches equilibrium when the voltage equals zero. Determine all time values when the circuit reaches equilibrium, including any complex solutions that represent the complete mathematical behavior of the system. Answer: t = 2, t = 1 + i, t = 1 - i Solution: Set up the equation V(t) = 0: t³ - 4t² + 6t - 4 = 0 Test possible rational roots using the Rational Root Theorem. Try t = 2: (2)³ - 4(2)² + 6(2) - 4 = 8 - 16 + 12 - 4 = 0. So t = 2 is a root.
   Full step-by-step solution

   Step 1: Set up the equation V(t) = 0: t³ - 4t² + 6t - 4 = 0 Step 2: Test possible rational roots using the Rational Root Theorem. Try t = 2: (2)³ - 4(2)² + 6(2) - 4 = 8 - 16 + 12 - 4 = 0. So t = 2 is a root. Step 3: Use synthetic division with t = 2: Coefficients: 1, -4, 6, -4 Bring down 1, multiply by 2: 2, add to -4: -2, multiply by 2: -4, add to 6: 2, multiply by 2: 4, add to -4: 0 Step 4: The quotient is t² - 2t + 2 Step 5: Solve t² - 2t + 2 = 0 using the quadratic formula: t = [2 ± sqrt(4 - 8)]/2 = [2 ± sqrt(-4)]/2 = [2 ± 2i]/2 = 1 ± i Step 6: The complete solution set is t = 2, t = 1 + i, t = 1 - i

4. An electrical engineer is designing a circuit where the voltage response is modeled by the polynomial V(t) = t³ - 4t² + 9t - 10, where t represents time in milliseconds. To ensure circuit stability, she needs to find all time values when the voltage equals zero, including any complex solutions that might indicate resonance frequencies. Find all solutions to V(t) = 0. Answer: 2, 1+2i, 1-2i Solution: Set up the equation: t³ - 4t² + 9t - 10 = 0 Use the Rational Root Theorem to test possible rational roots. Test t = 2: (2)³ - 4(2)² + 9(2) - 10 = 8 - 16 + 18 - 10 = 0. So t = 2 is a root.
   Full step-by-step solution

   Step 1: Set up the equation: t³ - 4t² + 9t - 10 = 0 Step 2: Use the Rational Root Theorem to test possible rational roots. Test t = 2: (2)³ - 4(2)² + 9(2) - 10 = 8 - 16 + 18 - 10 = 0. So t = 2 is a root. Step 3: Factor out (t - 2) using polynomial division or synthetic division. Synthetic division with 2: coefficients are 1, -4, 9, -10 Bring down 1, multiply by 2: 2, add to -4: -2, multiply by 2: -4, add to 9: 5, multiply by 2: 10, add to -10: 0. Step 4: The quotient is t² - 2t + 5. Step 5: Solve t² - 2t + 5 = 0 using the quadratic formula: t = [2 ± sqrt(4 - 20)]/2 = [2 ± sqrt(-16)]/2 = [2 ± 4i]/2 = 1 ± 2i. Step 6: The complete solution set is t = 2, t = 1 + 2i, t = 1 - 2i.

5. Noah is a quantum physicist studying the energy states of a particle in a potential well. The energy levels (in electron volts) of the particle are determined by the polynomial equation E^3 - 9E^2 + 33E - 65 = 0, where E represents the energy. The system has one real energy level that corresponds to a stable state, and two complex energy levels that represent virtual states in the mathematical model. Determine all energy levels (including complex ones) that satisfy the equation. Answer: 5, 2 + 3i, 2 - 3i Solution: Set the equation to zero: E^3 - 9E^2 + 33E - 65 = 0. Use the Rational Root Theorem. Possible rational roots are factors of 65: ±1, ±5, ±13, ±65.
   Full step-by-step solution

   Step 1: Set the equation to zero: E^3 - 9E^2 + 33E - 65 = 0. Step 2: Use the Rational Root Theorem. Possible rational roots are factors of 65: ±1, ±5, ±13, ±65. Step 3: Test E = 5: (5)^3 - 9(5)^2 + 33(5) - 65 = 125 - 225 + 165 - 65 = 0. So E = 5 is a root. Step 4: Perform synthetic division with root 5 on coefficients 1, -9, 33, -65: Bring down 1. Multiply 1 by 5 = 5, add to -9 gives -4. Multiply -4 by 5 = -20, add to 33 gives 13. Multiply 13 by 5 = 65, add to -65 gives 0. Step 5: The quotient is E^2 - 4E + 13 = 0. Step 6: Solve using the quadratic formula: E = [4 ± sqrt((-4)^2 - 4(1)(13))] / (2(1)) = [4 ± sqrt(16 - 52)] / 2 = [4 ± sqrt(-36)] / 2 = [4 ± 6i] / 2 = 2 ± 3i. Step 7: The solutions are E = 5, E = 2 + 3i, E = 2 - 3i. The answer is 5, 2 + 3i, 2 - 3i.

6. Find all complex solutions of x³ - 4x² + 9x - 10 = 0 Answer: 2, 1+2i, 1-2i Solution: Test possible rational roots using the Rational Root Theorem. Try x = 2: (2)³ - 4(2)² + 9(2) - 10 = 8 - 16 + 18 - 10 = 0. So x = 2 is a root.
   Full step-by-step solution

   Step 1: Test possible rational roots using the Rational Root Theorem. Try x = 2: (2)³ - 4(2)² + 9(2) - 10 = 8 - 16 + 18 - 10 = 0. So x = 2 is a root. Step 2: Use synthetic division with x = 2: 2 | 1 -4 9 -10 | 2 -4 10 -------------- 1 -2 5 0 Step 3: The quotient is x² - 2x + 5. Solve x² - 2x + 5 = 0 using the quadratic formula: x = [2 ± √(4 - 20)]/2 = [2 ± √(-16)]/2 = [2 ± 4i]/2 = 1 ± 2i Step 4: The three complex solutions are x = 2, x = 1 + 2i, and x = 1 - 2i