Polynomial Complex

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Polynomial Complex · Grade 12 · Worksheet 3

1. An electrical engineer is analyzing the voltage response in a circuit using 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 circuit reaches equilibrium, including any complex solutions that represent theoretical mathematical behavior. Answer: t = 1, t = 1, t = 2 Solution: In polynomial analysis, finding roots involves determining where the function equals zero. For cubic polynomials, there are always three solutions in the complex number system, though some may be repeated real roots.
   Full step-by-step solution

   In polynomial analysis, finding roots involves determining where the function equals zero. For cubic polynomials, there are always three solutions in the complex number system, though some may be repeated real roots. Engineers use these mathematical models to understand system behavior, where complex roots can indicate oscillatory characteristics or theoretical properties of the model that don't correspond to physical reality but help complete the mathematical understanding.

2. A complex polynomial function f(z) = z³ - 6z² + 13z - 10 is graphed on the complex plane. The graph intersects the real axis at points corresponding to the real roots of the polynomial. If one root is known to be 2, and the other two roots are complex conjugates, what is the sum of the imaginary parts of the complex conjugate roots? Answer: 0 Solution: f(z) = z³ - 6z² + 13z - 10 One root is 2, and the other two roots are complex conjugates. Since z = 2 is a root, we can divide f(z) by (z - 2).
   Full step-by-step solution

   We are given the polynomial: f(z) = z³ - 6z² + 13z - 10 One root is 2, and the other two roots are complex conjugates. --- **Step 1: Use the known root to factor the polynomial** Since z = 2 is a root, we can divide f(z) by (z - 2). Using synthetic division with 2: Coefficients: 1, -6, 13, -10 Bring down 1 → multiply by 2 → 2 → add to -6 → -4 Multiply -4 by 2 → -8 → add to 13 → 5 Multiply 5 by 2 → 10 → add to -10 → 0. So the quotient is: z² - 4z + 5. Thus: f(z) = (z - 2)(z² - 4z + 5) --- **Step 2: Find the other two roots** Solve z² - 4z + 5 = 0. Discriminant: Δ = (-4)² - 4×1×5 = 16 - 20 = -4. So: z = [4 ± sqrt(-4)] / 2 = [4 ± 2i] / 2 = 2 ± i. --- **Step 3: Identify the complex conjugate roots and their imaginary parts** The complex conjugate roots are: 2 + i and 2 - i. Imaginary parts: For 2 + i, imaginary part = 1. For 2 - i, imaginary part = -1. --- **Step 4: Sum the imaginary parts** 1 + (-1) = 0. --- **Final Answer:** 0

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. The circuit reaches resonance when the voltage equals zero. Determine all time values when resonance occurs, including any complex solutions that represent the complete mathematical behavior of the system. Answer: t = 2, t = 1 + i, t = 1 - i Solution: In electrical engineering and physics, complex roots of polynomial equations often represent oscillatory behavior or phase relationships in systems.
   Full step-by-step solution

   In electrical engineering and physics, complex roots of polynomial equations often represent oscillatory behavior or phase relationships in systems. When solving cubic equations, there is always at least one real root, and the remaining roots form a complex conjugate pair if they are not real. This mathematical structure reflects physical phenomena where systems can have both steady-state and oscillatory components.

4. Find all complex solutions of x⁴ + 4x² + 16 = 0 Answer: ±√3 ± i, ±i√3 ± 1 Solution: Let u = x², so the equation becomes u² + 4u + 16 = 0 Solve the quadratic using the quadratic formula: u = [-4 ± √(16 - 64)]/2 Simplify: u = [-4 ± √(-48)]/2 = [-4 ± 4i√3]/2 = -2 ± 2i√3 Now solve x² = -2 + 2i√3 and x² = -2 - 2i√3 For x² = -2 + 2i√3, convert to polar form: magnitude = √(4 + 12) =…
   Full step-by-step solution

   Step 1: Let u = x², so the equation becomes u² + 4u + 16 = 0 Step 2: Solve the quadratic using the quadratic formula: u = [-4 ± √(16 - 64)]/2 Step 3: Simplify: u = [-4 ± √(-48)]/2 = [-4 ± 4i√3]/2 = -2 ± 2i√3 Step 4: Now solve x² = -2 + 2i√3 and x² = -2 - 2i√3 Step 5: For x² = -2 + 2i√3, convert to polar form: magnitude = √(4 + 12) = 4, angle = 2π/3 Step 6: Square roots: x = ±2(cos(π/3) + i sin(π/3)) = ±(1 + i√3) Step 7: For x² = -2 - 2i√3, magnitude = 4, angle = 4π/3 Step 8: Square roots: x = ±2(cos(2π/3) + i sin(2π/3)) = ±(-1 + i√3) Step 9: The four complex solutions are: 1 + i√3, -1 - i√3, -1 + i√3, 1 - i√3

5. Find all complex solutions of x⁴ + 16 = 0 Answer: 2+2i, 2-2i, -2+2i, -2-2i Solution: Rewrite the equation as x⁴ = -16 Express -16 in polar form: -16 = 16(cos(π) + i sin(π)) Apply De Moivre's theorem to find the fourth roots: x = ∜16 [cos((π + 2kπ)/4) + i sin((π + 2kπ)/4)] for k = 0, 1, 2, 3 For k = 0: x = 2[cos(π/4) + i sin(π/4)] = 2(√2/2 + i√2/2) = √2 + i√2 For k = 1: x =…
   Full step-by-step solution

   Step 1: Rewrite the equation as x⁴ = -16 Step 2: Express -16 in polar form: -16 = 16(cos(π) + i sin(π)) Step 3: Apply De Moivre's theorem to find the fourth roots: x = ∜16 [cos((π + 2kπ)/4) + i sin((π + 2kπ)/4)] for k = 0, 1, 2, 3 Step 4: For k = 0: x = 2[cos(π/4) + i sin(π/4)] = 2(√2/2 + i√2/2) = √2 + i√2 Step 5: For k = 1: x = 2[cos(3π/4) + i sin(3π/4)] = 2(-√2/2 + i√2/2) = -√2 + i√2 Step 6: For k = 2: x = 2[cos(5π/4) + i sin(5π/4)] = 2(-√2/2 - i√2/2) = -√2 - i√2 Step 7: For k = 3: x = 2[cos(7π/4) + i sin(7π/4)] = 2(√2/2 - i√2/2) = √2 - i√2 Step 8: Simplify: √2 + i√2 = 2+2i, -√2 + i√2 = -2+2i, -√2 - i√2 = -2-2i, √2 - i√2 = 2-2i The solutions are 2+2i, 2-2i, -2+2i, -2-2i

6. A complex polynomial function f(z) = z³ - 5z² + 11z - 15 is graphed on the complex plane. The graph intersects the real axis at one point, and the other two roots form a complex conjugate pair. If you visualize these complex roots as vectors from the origin in the complex plane, what is the magnitude (distance from origin) of either complex root? Answer: √10 Solution: Find the real root by testing possible rational roots. Try z = 3: f(3) = 27 - 45 + 33 - 15 = 0, so z = 3 is a real root.
   Full step-by-step solution

   Step 1: Find the real root by testing possible rational roots. Try z = 3: f(3) = 27 - 45 + 33 - 15 = 0, so z = 3 is a real root. Step 2: Perform polynomial division: (z³ - 5z² + 11z - 15) ÷ (z - 3) Step 3: Using synthetic division with 3: coefficients 1, -5, 11, -15 Bring down 1, multiply by 3: 3, add to -5: -2, multiply by 3: -6, add to 11: 5, multiply by 3: 15, add to -15: 0 Step 4: The quotient is z² - 2z + 5 Step 5: Solve z² - 2z + 5 = 0 using quadratic formula: z = [2 ± √(4 - 20)]/2 = [2 ± √(-16)]/2 = [2 ± 4i]/2 = 1 ± 2i Step 6: The complex roots are 1 + 2i and 1 - 2i Step 7: Calculate magnitude: |1 + 2i| = √(1² + 2²) = √(1 + 4) = √5 Step 8: Verify: |1 - 2i| = √(1² + (-2)²) = √(1 + 4) = √5 Step 9: The magnitude of either complex root is √5.

7. Matiu is a marine biologist studying the population dynamics of a rare species of fish in a remote lake. He models the population size P(t) (in hundreds) over time t (in years) using the polynomial function P(t) = t³ - 9t² + 33t - 45. The population reaches a critical equilibrium point when P(t) = 0, which could indicate extinction or recovery thresholds. Determine all time values (including any complex solutions) when the population reaches equilibrium. Answer: t = 3, t = 3 + 2i, t = 3 - 2i Solution: Set the equation P(t) = 0: t³ - 9t² + 33t - 45 = 0. Use the Rational Root Theorem. Possible rational roots are factors of 45: ±1, ±3, ±5, ±9, ±15, ±45.
   Full step-by-step solution

   Step 1: Set the equation P(t) = 0: t³ - 9t² + 33t - 45 = 0. Step 2: Use the Rational Root Theorem. Possible rational roots are factors of 45: ±1, ±3, ±5, ±9, ±15, ±45. Step 3: Test t = 3: (3)³ - 9(3)² + 33(3) - 45 = 27 - 81 + 99 - 45 = 0. So t = 3 is a root. Step 4: Perform synthetic division with root 3. Coefficients: 1, -9, 33, -45. Bring down 1. Multiply 1 by 3 = 3, add to -9 = -6. Multiply -6 by 3 = -18, add to 33 = 15. Multiply 15 by 3 = 45, add to -45 = 0. The quotient is t² - 6t + 15. Step 5: Solve t² - 6t + 15 = 0 using the quadratic formula: t = [6 ± sqrt(36 - 60)] / 2 = [6 ± sqrt(-24)] / 2. Step 6: Simplify sqrt(-24) = sqrt(24) * i = 2*sqrt(6) * i. So t = [6 ± 2i*sqrt(6)] / 2 = 3 ± i*sqrt(6). Step 7: The complete solution set is t = 3, t = 3 + i*sqrt(6), t = 3 - i*sqrt(6).