Polynomial Complex

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Polynomial Complex · Grade 12 · Worksheet 1

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. Answer: 3, 2 + i, 2 - i Solution: Set Z(s) = 0: s³ - 7s² + 17s - 15 = 0. Use the Rational Root Theorem. Possible rational roots are factors of 15: ±1, ±3, ±5, ±15.
   Full step-by-step solution

   Step 1: Set Z(s) = 0: s³ - 7s² + 17s - 15 = 0. Step 2: Use the Rational Root Theorem. Possible rational roots are factors of 15: ±1, ±3, ±5, ±15. Step 3: Test s = 3: (3)³ - 7(3)² + 17(3) - 15 = 27 - 63 + 51 - 15 = 0. So s = 3 is a root. Step 4: Factor out (s - 3) using synthetic division with coefficients 1, -7, 17, -15: Bring down 1. Multiply 1 by 3: 3, add to -7: -4. Multiply -4 by 3: -12, add to 17: 5. Multiply 5 by 3: 15, add to -15: 0. The quotient is s² - 4s + 5. Step 5: Solve s² - 4s + 5 = 0 using the quadratic formula: s = [4 ± sqrt((-4)² - 4(1)(5))] / (2*1) = [4 ± sqrt(16 - 20)] / 2 = [4 ± sqrt(-4)] / 2 = [4 ± 2i] / 2 = 2 ± i. Step 6: The complete solution set is s = 3, s = 2 + i, s = 2 - i. The answer is 3, 2 + i, 2 - i.

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. Answer: t = 2, t = 3 + i, t = 3 - i Solution: Set the displacement function equal to zero: t^3 - 8t^2 + 22t - 20 = 0. Use the Rational Root Theorem. Possible rational roots are factors of 20: ±1, ±2, ±4, ±5, ±10, ±20.
   Full step-by-step solution

   Step 1: Set the displacement function equal to zero: t^3 - 8t^2 + 22t - 20 = 0. Step 2: Use the Rational Root Theorem. Possible rational roots are factors of 20: ±1, ±2, ±4, ±5, ±10, ±20. Step 3: Test t = 2: (2)^3 - 8(2)^2 + 22(2) - 20 = 8 - 32 + 44 - 20 = 0. So t = 2 is a root. Step 4: Use synthetic division to factor the polynomial. Coefficients: 1, -8, 22, -20. Bring down 1. Multiply by 2: 2, add to -8: -6. Multiply by 2: -12, add to 22: 10. Multiply by 2: 20, add to -20: 0. Step 5: The quotient is t^2 - 6t + 10. Step 6: Solve the quadratic t^2 - 6t + 10 = 0 using the quadratic formula: t = [6 ± sqrt((-6)^2 - 4*1*10)] / (2*1) = [6 ± sqrt(36 - 40)] / 2 = [6 ± sqrt(-4)] / 2 = [6 ± 2i] / 2 = 3 ± i. Step 7: The complete solution set is t = 2, t = 3 + i, t = 3 - i. The answer is t = 2, t = 3 + i, t = 3 - i.

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. Answer: t = 1, t = 1, t = 2 Solution: Set up the equation V(t) = 0: t³ - 4t² + 5t - 2 = 0 Use the Rational Root Theorem to test possible rational roots: ±1, ±2 Test t = 1: (1)³ - 4(1)² + 5(1) - 2 = 1 - 4 + 5 - 2 = 0, so t = 1 is a root Use synthetic division with t = 1: coefficients are 1, -4, 5, -2 Synthetic division: bring down 1,…
   Full step-by-step solution

   Step 1: Set up the equation V(t) = 0: t³ - 4t² + 5t - 2 = 0 Step 2: Use the Rational Root Theorem to test possible rational roots: ±1, ±2 Step 3: Test t = 1: (1)³ - 4(1)² + 5(1) - 2 = 1 - 4 + 5 - 2 = 0, so t = 1 is a root Step 4: Use synthetic division with t = 1: coefficients are 1, -4, 5, -2 Step 5: Synthetic division: bring down 1, multiply by 1 to get 1, add to -4 to get -3, multiply by 1 to get -3, add to 5 to get 2, multiply by 1 to get 2, add to -2 to get 0 Step 6: The quotient is t² - 3t + 2 Step 7: Factor the quadratic: t² - 3t + 2 = (t - 1)(t - 2) Step 8: The complete factorization is (t - 1)²(t - 2) = 0 Step 9: Set each factor equal to zero: t - 1 = 0 gives t = 1, and t - 2 = 0 gives t = 2 Step 10: The solutions are t = 1 (with multiplicity 2) and t = 2 Step 11: All solutions are real: t = 1, t = 1, t = 2 The answer is t = 1, t = 1, t = 2.

4. x³ - 12x² + 49x - 58 = 0 Answer: 2, 5 + 2i, 5 - 2i Solution: Test x = 2: (2)³ - 12(2)² + 49(2) - 58 = 8 - 48 + 98 - 58 = 0. So x = 2 is a root. Use synthetic division with 2 on coefficients 1, -12, 49, -58.
   Full step-by-step solution

   Step 1: Test x = 2: (2)³ - 12(2)² + 49(2) - 58 = 8 - 48 + 98 - 58 = 0. So x = 2 is a root. Step 2: Use synthetic division with 2 on coefficients 1, -12, 49, -58. Bring down 1, multiply by 2 to get 2, add to -12 to get -10, multiply by 2 to get -20, add to 49 to get 29, multiply by 2 to get 58, add to -58 to get 0. The quotient is x² - 10x + 29. Step 3: Solve x² - 10x + 29 = 0 using the quadratic formula: x = [10 ± sqrt(100 - 116)]/2 = [10 ± sqrt(-16)]/2 = [10 ± 4i]/2 = 5 ± 2i. Step 4: The three solutions are x = 2, x = 5 + 2i, and x = 5 - 2i.

5. A complex polynomial function f(z) = z³ - 3z² + 4z - 2 has one real root at z = 1. On the complex plane, this function's graph intersects the imaginary axis. If you visualize the polynomial's behavior, what are the other two complex roots? Answer: 1 ± i Solution: We are given: f(z) = z³ - 3z² + 4z - 2, and one real root is z = 1. Substitute z = 1: f(1) = 1 - 3 + 4 - 2 = (1 - 3) = -2, then -2 + 4 = 2, then 2 - 2 = 0. Yes, f(1) = 0.
   Full step-by-step solution

   We are given: f(z) = z³ - 3z² + 4z - 2, and one real root is z = 1. Step 1: Verify z = 1 is a root. Substitute z = 1: f(1) = 1 - 3 + 4 - 2 = (1 - 3) = -2, then -2 + 4 = 2, then 2 - 2 = 0. Yes, f(1) = 0. Step 2: Factor out (z - 1). We perform polynomial division: Divide z³ - 3z² + 4z - 2 by (z - 1). Using synthetic division with 1: Coefficients: 1 (z³), -3 (z²), 4 (z), -2 (constant) Bring down 1. Multiply 1 * 1 = 1, add to -3: -3 + 1 = -2. Multiply 1 * (-2) = -2, add to 4: 4 - 2 = 2. Multiply 1 * 2 = 2, add to -2: -2 + 2 = 0. So quotient is: z² - 2z + 2. Thus f(z) = (z - 1)(z² - 2z + 2). Step 3: Solve z² - 2z + 2 = 0. Use the quadratic formula: z = [2 ± sqrt(4 - 8)] / 2 = [2 ± sqrt(-4)] / 2 = [2 ± 2i] / 2 = 1 ± i. Step 4: Conclusion. The other two complex roots are 1 + i and 1 - i. Final answer: 1 ± i

6. A complex polynomial function f(z) = z³ - 3z² + 4z - 2 is graphed on the complex plane. The function has one real root at z = 1. Find the other two complex roots, expressing your answer in the form a ± bi. Answer: 1 ± i Solution: We are given: f(z) = z³ - 3z² + 4z - 2, and one real root is z = 1. Substitute z = 1: f(1) = 1³ - 3(1)² + 4(1) - 2 = 1 - 3 + 4 - 2 = 0. Yes, it is a root.
   Full step-by-step solution

   We are given: f(z) = z³ - 3z² + 4z - 2, and one real root is z = 1. Step 1: Verify z = 1 is a root. Substitute z = 1: f(1) = 1³ - 3(1)² + 4(1) - 2 = 1 - 3 + 4 - 2 = 0. Yes, it is a root. Step 2: Divide the cubic polynomial by (z - 1). We use polynomial long division or synthetic division. Synthetic division with 1: Coefficients: 1 (z³), -3 (z²), 4 (z), -2 (constant) Bring down 1. Multiply 1 * 1 = 1, add to -3: -3 + 1 = -2. Multiply 1 * (-2) = -2, add to 4: 4 - 2 = 2. Multiply 1 * 2 = 2, add to -2: -2 + 2 = 0. Quotient coefficients: 1, -2, 2. So f(z) = (z - 1)(z² - 2z + 2). Step 3: Solve z² - 2z + 2 = 0. Use quadratic formula: z = [2 ± sqrt(4 - 8)] / 2 = [2 ± sqrt(-4)] / 2 = [2 ± 2i] / 2 = 1 ± i. Step 4: Conclusion. The other two roots are 1 + i and 1 - i. Final answer: 1 ± i

7. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream using the polynomial function C(t) = t³ - 6t² + 11t - 6, where t represents hours after administration and C(t) is the concentration in milligrams per liter. The medication becomes ineffective when the concentration reaches zero. At what times after administration will the medication become ineffective? Answer: 1, 2, 3 hours Solution: C(t) = t^3 - 6t^2 + 11t - 6 The medication becomes ineffective when C(t) = 0. t^3 - 6t^2 + 11t - 6 = 0 Possible rational roots are factors of the constant term (-6): ±1, ±2, ±3, ±6.
   Full step-by-step solution

   We are given the concentration function: C(t) = t^3 - 6t^2 + 11t - 6 The medication becomes ineffective when C(t) = 0. So we solve: t^3 - 6t^2 + 11t - 6 = 0 --- **Step 1: Try possible rational roots** Possible rational roots are factors of the constant term (-6): ±1, ±2, ±3, ±6. Test t = 1: (1)^3 - 6*(1)^2 + 11*(1) - 6 = 1 - 6 + 11 - 6 = 0 So t = 1 is a root. --- **Step 2: Polynomial division** Divide t^3 - 6t^2 + 11t - 6 by (t - 1): Using synthetic division with 1: Coefficients: 1, -6, 11, -6 Bring down 1 → multiply by 1 → 1, add to -6 → -5 → multiply by 1 → -5, add to 11 → 6 → multiply by 1 → 6, add to -6 → 0. Quotient: t^2 - 5t + 6. So: t^3 - 6t^2 + 11t - 6 = (t - 1)(t^2 - 5t + 6) --- **Step 3: Factor the quadratic** t^2 - 5t + 6 = (t - 2)(t - 3) Check: t^2 - 2t - 3t + 6 = t^2 - 5t + 6, correct. --- **Step 4: Write fully factored form** C(t) = (t - 1)(t - 2)(t - 3) --- **Step 5: Find roots** Set C(t) = 0: (t - 1)(t - 2)(t - 3) = 0 So t = 1, t = 2, t = 3. --- **Step 6: Interpret** These are the times (in hours) after administration when the concentration is zero, meaning the medication becomes ineffective. Thus, the medication becomes ineffective at **1 hour, 2 hours, and 3 hours** after administration.