please send handwritten solution step by step Q6 part b Q6In what follows, let i = v-1 denote the imaginary unit.(a) It z = 4 +5i and w = 2+4i, find32 + 4w3z - 4wShow the steps. Express your answer using integers or rationals for the real and imaginary parts.(b) Given that x = 1- 4i is one root of the quartic equation2*- 82' + 54z – 152z + 425 = 0,tind the other three roots. Show your working in detail.(c) Use de Moivre's formula to write down a formula for all of the n-th roots of z = rexp(ie). In otherwords, give a formula for the complex numbers w, such that (w)" =z, wherek = 0, 1, 2,...,n– 1. Use the formula to caiculate alli of the 4-th roots of the complex number2= 16exp().You must show all of your working and must express each root in the representation:w = R(cos(4,7) + i sin(,7)],where Rand o, should be expressed in simplified form (e.g., using rationals and/or roots), not asdecimals. If the roots, w, were plotted on an Argand diagram, what type of geometrical figure wouldbe formed? Draw this figure, labelling the roots.(d) Given any complex number, z, let ztn denote the repeated power:if n =0(2 1 (n - 1)) ifn 1.'so that n counts the number of exponentiations. For example,310 = 3, 311= (3f 0)* = 3* = 27, 312= (31 1) = (3') = 27* = 19683.Calculate i ↑ 0, i f 1, i ↑ 2, i 1 3, and i T 4. Hence deduce a general formula for i ↑n for n 0(you may find it helptul to write n = 4m + k tfor m 0 and k = 0, 1,2, 3). Show all working. ALPHABETIn all questions, you should work with an alphabet of n = 37 symbols, as indicated below.For individual symbols, you should use integer codes in the range 0 to 36 inclusive (therefore, you will bemainly using arithmetic modulo n = 37), as indicated by the following table:AвDEFG0 1JK234578LMO P14 15Q|RS10 11121316171819TU20 21V.WY?222324 25 26 272829@30 3132 3334 35 36So, for example, the symbol C should be encoded as the integer 3.Note: the space character (used to separate words) should be encoded by the integer 0.

Question

please send handwritten solution step by step Q6 part b Q6In what follows, let i = v-1 denote the imaginary unit.(a) It z = 4 +5i and w = 2+4i, find32 + 4w3z - 4wShow the steps. Express your answer using integers or rationals for the real and imaginary parts.(b) Given that x = 1- 4i is one root of the quartic equation2*- 82' + 54z – 152z + 425 = 0,tind the other three roots. Show your working in detail.(c) Use de Moivre's formula to write down a formula for all of the n-th roots of z = rexp(ie). In otherwords, give a formula for the complex numbers w, such that (w)&#34; =z, wherek = 0, 1, 2,...,n– 1. Use the formula to caiculate alli of the 4-th roots of the complex number2= 16exp().You must show all of your working and must express each root in the representation:w = R(cos(4,7) + i sin(,7)],where Rand o, should be expressed in simplified form (e.g., using rationals and/or roots), not asdecimals. If the roots, w, were plotted on an Argand diagram, what type of geometrical figure wouldbe formed? Draw this figure, labelling the roots.(d) Given any complex number, z, let ztn denote the repeated power:if n =0(2 1 (n - 1)) ifn >1.'so that n counts the number of exponentiations. For example,310 = 3, 311= (3f 0)* = 3* = 27, 312= (31 1) = (3') = 27* = 19683.Calculate i ↑ 0, i f 1, i ↑ 2, i 1 3, and i T 4. Hence deduce a general formula for i ↑n for n >0(you may find it helptul to write n = 4m + k tfor m > 0 and k = 0, 1,2, 3). Show all working. ALPHABETIn all questions, you should work with an alphabet of n = 37 symbols, as indicated below.For individual symbols, you should use integer codes in the range 0 to 36 inclusive (therefore, you will bemainly using arithmetic modulo n = 37), as indicated by the following table:AвDEFG0 1JK234578LMO P14 15Q|RS10 11121316171819TU20 21V.WY?222324 25 26 272829@30 3132 3334 35 36So, for example, the symbol C should be encoded as the integer 3.Note: the space character (used to separate words) should be encoded by the integer 0.

Accepted Answer

We will use long division to factor the four degree polynomial into the product of two quadratic…

Given that x = 1-4i is one root of the quartic equation 2-82' + 54z - 152z +425 = 0, find the other three roots. Show your working in detail.