1. Show that the absolute value of Re(z) + the absolute value of Im(z) is less or equal to root of 2 absolute value of z for any z is in complex number. 2. Under what circumstances does the absolute value of z + w= the absolute value of z + the absolute value of w where z;w in C. 3. Determine if the following is true or false with reason: Arg(z)=-Arg(z) if z is not real number, where Arg(z) denotes the principal argument of z.

Q: 4. Given that 3-2i is a root of the equation z-6z³+18z²+pz+65-0, find all the roots and the real…

A: If  is a root of  then its conjugate  is also a root of  and hence  is a factor of .

Q: Prove this theorem Let a,b,c be an element of real numbers where a is not equal to zero. The…

A: Use the provided information.

Q: Lim. Zan² (4x) X² it: to x 70

A: Let,the given limit is

Q: show IBat = (at 5) (1+i) (a+bi where 9,6 Are reals Hense tEs Find the value 2 (9-31) (a43i) 15-51)

A:

Q: 4. Consider the complex number w = 3cis (5). For which value(s) of k is x – wa factor of the…

A: I am going to solve the given problem by using some simple geometry to get the required result.

Q: w is an imaginary root of unity. Prove that (i) (a+ba+ca²)³ + (a+ba² +ca)³ = (2a-b-c) (2b-a-c)…

A:

Q: 2 Express ( √5 - √5₂) in 2 simplest a + bi form.

A:

Q: a. Factor the polynomial over the set of real numbers. b. Factor the polynomial over the set of…

A: 107. Given polynomial is: fx=x4+2x3+x2+8x-12 We have to factor the above polynomial over the set of…

Q: 20. Express (-3 - 8i) (-7 + 2i) + (7 - 3i) in simplest a + bi form

A:

Q: Let Describe all solutions of Ax = x = x₂ +x3 +x4 1 -₁2 50]. 4 -5 -10 8 10 A =

A:

Q: 59. Let a, c +0. Show that the roots of ax? + bx +c = 0 cx? + bx + a =0 and are reciprocals of each…

A:

Q: b) Let {(-2,–6), (–1,5),(0,4),(1,3)} and {(-2,–18), (-1,–7),(0,–4), (1,–3)} be the point-value…

A:

Q: Which set of values for x should be tested to determine the possible zeros of 2x³ - 3x² + 3x - 10? O…

A:

Q: a) Calculate the 4th roots of (5-j3)

A: The objective is to calculate the 4th roots of (5-j 3).

Q: Use the equation y = x² - 17x + 72 to answer the following question. For what values of x is y ≥ 0?

A:

Q: 2 The roots of the equation X + 2x+ 5 0 a 1) -3 and 1 2) -1, only 3)-1+ 2i and 1- 2i 4) -1+4i and…

A: given equation :

Q: Prove that zª +1 = (r² + √2x+1)(x² − √√2+1). Then factor z¹+1 into linear factors over the complex…

A:

Q: Prove that the roots of ax+4=2x²-bx are real for all values of a and b

A: The nature of roots of a quadratic equation Ax2+Bx+C=0  is:

Q: 2. Show that Re(iz)=-Im z. 3. Show that Re(iz)=-Im z.

A:

Q: Show that the first statement is true while the second statement is false. There exist real numbers…

A:

Q: 2 Express (-i)² i 3 3 in 2 simplest a + bi form.

A:

Q: Simplify and express in a + bi form: (12 + 3i) – (3 – i) 2.

A: 2. We need to simplify the following complex number expression 12+3i - 3-i

Q: 10. Use i = (0, 1) and y = (y, 0) to verify that -(iy) = (-i)y. Thus show that the addi- tive…

A: i=0 ,1y=y ,0 verify :   -iy = -iy  To find additive inverse of z=x+iy

Q: Following the definition of the principal argument of the complex number and the fact that Arg(z1…

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.