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 - 3t + 2), where t represents hours after administration. The drug becomes ineffective when its concentration drops below detectable levels. At what time does the concentration approach zero as t increases indefinitely? Answer: The concentration approaches 0 as t approaches infinity Solution: For rational functions where the degree of the numerator is greater than the degree of the denominator, the function will not have a finite horizontal asymptote.
   Full step-by-step solution

   For rational functions where the degree of the numerator is greater than the degree of the denominator, the function will not have a finite horizontal asymptote. Instead, as the input variable increases without bound, the function values will either increase or decrease without bound. The end behavior of such functions is determined by comparing the leading terms of the numerator and denominator polynomials.

2. 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.

3. 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.

4. A complex polynomial function f(z) = z^4 - 4z^3 + 14z^2 - 20z + 25 is graphed on the complex plane. The graph shows two pairs of complex conjugate roots. If one root is known to be 1 + 2i, what is the product of all four roots of this polynomial? Answer: 25 Solution: For any polynomial of the form z^n + a_{n-1}z^{n-1} + ... + a_0, the product of all roots equals (-1)^n * a_0. Our polynomial is z^4 - 4z^3 + 14z^2 - 20z + 25, which is monic (leading coefficient is 1).
   Full step-by-step solution

   Step 1: For any polynomial of the form z^n + a_{n-1}z^{n-1} + ... + a_0, the product of all roots equals (-1)^n * a_0. Step 2: Our polynomial is z^4 - 4z^3 + 14z^2 - 20z + 25, which is monic (leading coefficient is 1). Step 3: Here n = 4 and a_0 = 25. Step 4: The product of all roots = (-1)^4 * 25 = 1 * 25 = 25. Step 5: Therefore, regardless of the specific roots, the product of all four roots is 25. The answer is 25.

5. Ava, a theoretical physicist, is studying the quantum behavior of a particle in a potential well. The allowed energy states E (in electronvolts) of the particle are determined by the characteristic polynomial equation E^3 - 8E^2 + 22E - 20 = 0. The system has one real energy level that corresponds to a stable bound state, and two complex energy levels that represent virtual states in the mathematical model used to describe tunneling effects. Determine all energy levels (including complex ones) that satisfy the equation. Answer: 2, 3 + i, 3 - i Solution: Set the equation to zero: E^3 - 8E^2 + 22E - 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 equation to zero: E^3 - 8E^2 + 22E - 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 E = 2: (2)^3 - 8(2)^2 + 22(2) - 20 = 8 - 32 + 44 - 20 = 0. So E = 2 is a root. Step 4: Perform synthetic division with root 2 on coefficients 1, -8, 22, -20: Bring down 1. Multiply 1 by 2 = 2, add to -8 gives -6. Multiply -6 by 2 = -12, add to 22 gives 10. Multiply 10 by 2 = 20, add to -20 gives 0. Step 5: The quotient is E^2 - 6E + 10 = 0. Step 6: Solve using the quadratic formula: E = [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 solutions are E = 2, E = 3 + i, E = 3 - i. The answer is 2, 3 + i, 3 - i.

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. 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.