Polynomial Complex

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Polynomial Complex · Grade 12 · Worksheet 1

1. 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. Answer: 2+i, 2-i, 2+i, 2-i Solution: In polynomial modeling of physical systems, complex roots often appear in conjugate pairs and indicate oscillatory behavior in the mathematical model. The fundamental theorem of algebra guarantees that a degree 4 polynomial will have exactly 4 roots when counting multiplicity.
   Full step-by-step solution

   In polynomial modeling of physical systems, complex roots often appear in conjugate pairs and indicate oscillatory behavior in the mathematical model. For quartic polynomials, factoring techniques or recognizing special patterns can help find all roots. The fundamental theorem of algebra guarantees that a degree 4 polynomial will have exactly 4 roots when counting multiplicity.

2. Find all complex solutions of x³ - 2x² + 4x - 8 = 0 Answer: 2, 2i, -2i Solution: Factor by grouping: (x³ - 2x²) + (4x - 8) = x²(x - 2) + 4(x - 2) = (x - 2)(x² + 4) Set each factor equal to zero: x - 2 = 0 or x² + 4 = 0 Solve x - 2 = 0: x = 2 Solve x² + 4 = 0: x² = -4, x = ±√(-4) = ±2i The three complex solutions are x = 2, x = 2i, and x = -2i
   Full step-by-step solution

   Step 1: Factor by grouping: (x³ - 2x²) + (4x - 8) = x²(x - 2) + 4(x - 2) = (x - 2)(x² + 4) Step 2: Set each factor equal to zero: x - 2 = 0 or x² + 4 = 0 Step 3: Solve x - 2 = 0: x = 2 Step 4: Solve x² + 4 = 0: x² = -4, x = ±√(-4) = ±2i Step 5: The three complex solutions are x = 2, x = 2i, and x = -2i

3. x³ - 3x² + 9x - 27 = 0 Answer: 3, 3i, -3i Solution: Test x = 3: (3)³ - 3(3)² + 9(3) - 27 = 27 - 27 + 27 - 27 = 0, so x = 3 is a root. Use synthetic division with 3. Coefficients: 1, -3, 9, -27.
   Full step-by-step solution

   Step 1: Test x = 3: (3)³ - 3(3)² + 9(3) - 27 = 27 - 27 + 27 - 27 = 0, so x = 3 is a root. Step 2: Use synthetic division with 3. Coefficients: 1, -3, 9, -27. Bring down 1. Multiply 1 by 3 to get 3, add to -3 to get 0. Multiply 0 by 3 to get 0, add to 9 to get 9. Multiply 9 by 3 to get 27, add to -27 to get 0. The quotient is x² + 0x + 9 = x² + 9. Step 3: Solve x² + 9 = 0. Subtract 9: x² = -9. Take square root: x = ± sqrt(-9) = ± 3i. Step 4: The three solutions are x = 3, x = 3i, and x = -3i.

4. Sophia, an aerospace engineer, is analyzing the stability of a satellite's orbit. The orbital perturbation is modeled by the polynomial equation x³ - 6x² + 21x - 26 = 0, where x represents a dimensionless stability parameter. Find all solutions (real and complex) to this equation. Answer: 2, 2+3i, 2-3i Solution: Solve x³ - 6x² + 21x - 26 = 0. Use the Rational Root Theorem: possible rational roots are ±1, ±2, ±13, ±26. Test x = 2: (2)³ - 6(2)² + 21(2) - 26 = 8 - 24 + 42 - 26 = 0.
   Full step-by-step solution

   Step 1: Solve x³ - 6x² + 21x - 26 = 0. Use the Rational Root Theorem: possible rational roots are ±1, ±2, ±13, ±26. Test x = 2: (2)³ - 6(2)² + 21(2) - 26 = 8 - 24 + 42 - 26 = 0. So x = 2 is a root. Step 2: Factor out (x - 2) using synthetic division. Coefficients: 1, -6, 21, -26. Bring down 1. Multiply 1 by 2: 2, add to -6: -4. Multiply -4 by 2: -8, add to 21: 13. Multiply 13 by 2: 26, add to -26: 0. The quotient is x² - 4x + 13. Step 3: Solve x² - 4x + 13 = 0 using the quadratic formula: x = [4 ± sqrt((-4)² - 4(1)(13))] / (2(1)) = [4 ± sqrt(16 - 52)] / 2 = [4 ± sqrt(-36)] / 2 = [4 ± 6i] / 2 = 2 ± 3i. Step 4: The solutions are x = 2, x = 2 + 3i, x = 2 - 3i. The answer is 2, 2+3i, 2-3i.

5. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (t³ - 6t² + 11t - 6)/(t² - 3t + 2), where t represents hours after administration. The function appears to be undefined at certain time values. Determine all complex time values where the medication concentration would be mathematically undefined, and express your answer in the form a ± bi. Answer: 1 ± i Solution: While some of these points correspond to real values where the model fails, others may be complex numbers that don't correspond to measurable physical quantities but are mathematically significant.
   Full step-by-step solution

   In mathematical modeling of real-world phenomena, rational functions can have points where they are undefined due to division by zero. While some of these points correspond to real values where the model fails, others may be complex numbers that don't correspond to measurable physical quantities but are mathematically significant. Understanding the complete set of solutions, including complex ones, provides insight into the full mathematical structure of the model.

6. 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 time in hours and C(t) represents concentration in milligrams per liter. The medication becomes ineffective when the concentration reaches zero. At what times will the medication need to be readministered to maintain therapeutic levels? Answer: 1, 2, and 3 hours Solution: C(t) = t³ - 6t² + 11t - 6 The medication becomes ineffective when C(t) = 0. t³ - 6t² + 11t - 6 = 0 Look for an integer root using the Rational Root Theorem.
   Full step-by-step solution

   We are given the concentration function: C(t) = t³ - 6t² + 11t - 6 The medication becomes ineffective when C(t) = 0. So we solve: t³ - 6t² + 11t - 6 = 0 --- **Step 1: Look for an integer root using the Rational Root Theorem.** Possible rational roots: factors of 6 divided by factors of 1 → ±1, ±2, ±3, ±6. Test t = 1: 1³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 Yes, t = 1 is a root. --- **Step 2: Factor out (t - 1) using polynomial division or synthetic division.** 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: t² - 5t + 6 So: t³ - 6t² + 11t - 6 = (t - 1)(t² - 5t + 6) --- **Step 3: Factor the quadratic.** t² - 5t + 6 = (t - 2)(t - 3) Thus: C(t) = (t - 1)(t - 2)(t - 3) --- **Step 4: Solve C(t) = 0.** (t - 1)(t - 2)(t - 3) = 0 So t = 1, t = 2, t = 3. --- **Step 5: Interpret the result.** The concentration is zero at t = 1 hour, t = 2 hours, and t = 3 hours after administration. So the medication needs to be readministered at these times to maintain therapeutic levels. --- **Final answer:** 1, 2, and 3 hours

7. x³ - 2x² + 4x - 8 = 0 Answer: 2, 2i, -2i Solution: Factor by grouping: (x³ - 2x²) + (4x - 8) = 0 Factor each group: x²(x - 2) + 4(x - 2) = 0 Factor out (x - 2): (x - 2)(x² + 4) = 0 Set each factor equal to zero: x - 2 = 0 or x² + 4 = 0 Solve x - 2 = 0: x = 2 Solve x² + 4 = 0: x² = -4 Take square root of both sides: x = ±√(-4) Simplify: x = ±2i…
   Full step-by-step solution

   Step 1: Factor by grouping: (x³ - 2x²) + (4x - 8) = 0 Step 2: Factor each group: x²(x - 2) + 4(x - 2) = 0 Step 3: Factor out (x - 2): (x - 2)(x² + 4) = 0 Step 4: Set each factor equal to zero: x - 2 = 0 or x² + 4 = 0 Step 5: Solve x - 2 = 0: x = 2 Step 6: Solve x² + 4 = 0: x² = -4 Step 7: Take square root of both sides: x = ±√(-4) Step 8: Simplify: x = ±2i Step 9: The solutions are x = 2, x = 2i, and x = -2i