Polynomial Complex

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Polynomial Complex · Grade 12 · Worksheet 3

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? Answer: 5 Solution: The polynomial is f(z) = z³ - 9z² + 33z - 65. Since it has real coefficients, complex roots occur in conjugate pairs. One root is given as 3 + 4i.
   Full step-by-step solution

   Step 1: The polynomial is f(z) = z³ - 9z² + 33z - 65. Since it has real coefficients, complex roots occur in conjugate pairs. One root is given as 3 + 4i. Its conjugate is 3 - 4i. These form one pair. Step 2: Let the real root be r. The three roots are: r, 3 + 4i, 3 - 4i. Step 3: For a monic cubic (leading coefficient 1), the sum of all roots = -(coefficient of z²) = -(-9) = 9. Sum = r + (3 + 4i) + (3 - 4i) = r + 6 = 9, so r = 3. Step 4: The other complex root in the conjugate pair is 3 - 4i. Its magnitude = sqrt(3² + (-4)²) = sqrt(9 + 16) = sqrt(25) = 5. The answer is 5.

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. Answer: 3, 3+2i, 3-2i Solution: Set the polynomial equal to zero: t³ - 9t² + 33t - 45 = 0 Use the Rational Root Theorem. Possible rational roots are factors of 45: ±1, ±3, ±5, ±9, ±15, ±45. Test t = 3: (3)³ - 9(3)² + 33(3) - 45 = 27 - 81 + 99 - 45 = 0.
   Full step-by-step solution

   Step 1: Set the polynomial equal to zero: 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: Use synthetic division with root 3. Coefficients: 1, -9, 33, -45. Bring down 1. Multiply 1 by 3 = 3, add to -9 to get -6. Multiply -6 by 3 = -18, add to 33 to get 15. Multiply 15 by 3 = 45, add to -45 to get 0. The quotient is t² - 6t + 15. Step 5: Solve t² - 6t + 15 = 0 using the quadratic formula: t = [6 ± sqrt((-6)² - 4(1)(15))] / (2(1)) = [6 ± sqrt(36 - 60)] / 2 = [6 ± sqrt(-24)] / 2 Step 6: Simplify sqrt(-24) = sqrt(24) * i = sqrt(4 * 6) * i = 2*sqrt(6)*i. So t = [6 ± 2*sqrt(6)*i] / 2 = 3 ± sqrt(6)*i. Step 7: The complete solution set is t = 3, t = 3 + sqrt(6)*i, t = 3 - sqrt(6)*i. The answer is 3, 3+sqrt(6)*i, 3-sqrt(6)*i.

3. Find all complex solutions of x³ - 4x² + 9x - 10 = 0 given that 2 + i is a root. Answer: 2 + i, 2 - i, 1 Solution: Since 2 + i is a root and the polynomial has real coefficients, 2 - i must also be a root. The complex solutions are 2 + i, 2 - i, and 2.
   Full step-by-step solution

   Step 1: Since 2 + i is a root and the polynomial has real coefficients, 2 - i must also be a root. Step 2: The quadratic factor with roots 2 + i and 2 - i is: (x - (2 + i))(x - (2 - i)) = x² - 4x + 5 Step 3: Divide the original polynomial x³ - 4x² + 9x - 10 by x² - 4x + 5 Step 4: Polynomial division: (x³ - 4x² + 9x - 10) ÷ (x² - 4x + 5) = x - 2 Step 5: The remaining factor is x - 2 = 0, so x = 2 Step 6: The three roots are: 2 + i, 2 - i, and 2 Step 7: Verify: (2 + i) + (2 - i) + 2 = 6, which matches the negative of the coefficient of x² divided by the leading coefficient. The complex solutions are 2 + i, 2 - i, and 2.

4. Emma is a telecommunications engineer analyzing signal propagation in a fiber optic cable. The signal strength S(t) in decibels at time t seconds is modeled by the polynomial S(t) = t³ - 5t² + 11t - 15. The signal experiences a critical attenuation when S(t) = 0. Determine all time values (including complex solutions) when the signal strength reaches zero, expressing any non-real solutions in the form a ± bi. Answer: 3, 1 + 2i, 1 - 2i Solution: Set the polynomial equal to zero: t³ - 5t² + 11t - 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 the polynomial equal to zero: t³ - 5t² + 11t - 15 = 0. Step 2: Use the Rational Root Theorem. Possible rational roots are factors of -15: ±1, ±3, ±5, ±15. Step 3: Test t = 3: (3)³ - 5(3)² + 11(3) - 15 = 27 - 45 + 33 - 15 = 0. So t = 3 is a root. Step 4: Factor out (t - 3) using synthetic division. Coefficients: 1, -5, 11, -15. Step 5: 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 6: The quotient is t² - 2t + 5. Step 7: 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 8: The complete solution set is t = 3, t = 1 + 2i, t = 1 - 2i. The answer is 3, 1+2i, 1-2i.

5. Find all complex solutions of x⁴ - 16 = 0 Answer: 2, -2, 2i, -2i Solution: Start with the equation x⁴ - 16 = 0 Factor as a difference of squares: (x² - 4)(x² + 4) = 0 Set each factor equal to zero: x² - 4 = 0 or x² + 4 = 0 Solve x² - 4 = 0: x² = 4, so x = 2 or x = -2 Solve x² + 4 = 0: x² = -4, so x = 2i or x = -2i The complete solution set is {2, -2, 2i, -2i}
   Full step-by-step solution

   Step 1: Start with the equation x⁴ - 16 = 0 Step 2: Factor as a difference of squares: (x² - 4)(x² + 4) = 0 Step 3: Set each factor equal to zero: x² - 4 = 0 or x² + 4 = 0 Step 4: Solve x² - 4 = 0: x² = 4, so x = 2 or x = -2 Step 5: Solve x² + 4 = 0: x² = -4, so x = 2i or x = -2i Step 6: The complete solution set is {2, -2, 2i, -2i}

6. 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

7. 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