Math Problem Statement

Exercise 03 involves solving a polynomial equation and analyzing properties of complex numbers.

Solution

The uploaded file contains a math exercise on complex numbers and polynomials. Here's the solution process broken down for the given problem.


I. On the polynomial P(z)=z3(2+8i)z2(11+18i)z+2862iP(z) = z^3 - (2 + 8i)z^2 - (11 + 18i)z + 28 - 62i

1. Calculate P(2i)P(2 - i):

Substitute z=2iz = 2 - i into P(z)P(z): P(2i)=(2i)3(2+8i)(2i)2(11+18i)(2i)+2862iP(2 - i) = (2 - i)^3 - (2 + 8i)(2 - i)^2 - (11 + 18i)(2 - i) + 28 - 62i Expand each term to compute P(2i)P(2 - i).


2. Show P(z)=0P(z) = 0 admits a purely imaginary solution, and determine it.

Solve P(z)=0P(z) = 0, noting that at least one root is purely imaginary. Let z=biz = bi (where bRb \in \mathbb{R}) and substitute into P(z)P(z). Simplify to find bb.


3. Determine Q(z)Q(z) such that P(z)=(z+32i)Q(z)P(z) = (z + 3 - 2i)Q(z).

Perform synthetic or polynomial division of P(z)P(z) by (z+32i)(z + 3 - 2i) to find Q(z)Q(z).


4. Deduce the solutions of P(z)=0P(z) = 0.

Using P(z)=(z+32i)Q(z)P(z) = (z + 3 - 2i)Q(z), solve Q(z)=0Q(z) = 0 to find the remaining roots.


II. Points A,B,CA, B, C with affixes zA=3+2iz_A = -3 + 2i, zB=2iz_B = -2i, zC=5+8iz_C = 5 + 8i

1. Let Z=zAzCzAzBZ = \frac{z_A - z_C}{z_A - z_B}:

  • (a) Write ZZ in trigonometric form: Compute zAzCz_A - z_C and zAzBz_A - z_B, then express their ratio in modulus-argument form.

  • (b) Study the nature of triangle ABCABC: Use the relative lengths and angles of the points to classify ABCABC as isosceles, equilateral, or scalene, and determine if it is right-angled.


2. For Z=z+32iz+2iZ' = \frac{z + 3 - 2i}{z + 2i}:

  • (a) Find the set Ψ\Psi of points M(z)M(z) such that Z1=0|Z'| - 1 = 0: This condition implies Z=1|Z'| = 1, which geometrically represents a circle in the complex plane. Solve Z=1|Z'| = 1 to find the equation of the circle.

  • (b) Find the set Γ\Gamma of points M(z)M(z) such that ZZ' is purely imaginary: Solve Re(Z)=0\text{Re}(Z') = 0 to derive a geometric condition.


3. Prove that nN,(zB)4n=16n\forall n \in \mathbb{N}^*, (z_B)^{4n} = 16^n:

Using zB=2iz_B = -2i, calculate zB4z_B^4, and generalize for zB4nz_B^{4n}.


Would you like detailed step-by-step calculations for any specific part?

5 Related Questions

  1. What are the roots of P(z)P(z) if it is fully factored?
  2. How is a polynomial division performed in the complex plane?
  3. What is the geometric interpretation of a complex ratio in trigonometric form?
  4. How do you prove conditions for points on a circle in the complex plane?
  5. How does zBz_B behave under exponentiation, and what patterns can be generalized?

Tip

For polynomials with complex coefficients, symmetry in roots often simplifies solving; consider conjugate root theorems.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polynomials
Geometry in the Complex Plane

Formulas

Modulus and argument of complex numbers
Polynomial division
Conditions for purely imaginary numbers
Geometric properties of complex ratios

Theorems

Fundamental Theorem of Algebra
Conjugate Root Theorem
Properties of exponents for imaginary numbers

Suitable Grade Level

Grades 11-12 or College Level