The complex number z = 2 + i is the root of the polynomial z^4 – 6z^3 + 16z^2 - 22z + 15 = 0. Find the remaining roots c) Let z= √(3 - i) i) Plot z on an Argand diagram. ii) Let w = az where a > 0, a E R. Express w in polar form iii) Express w^8 in the form ka^n(x + i√y) where k,x,y E Z
since 2+i is a root, so is 2-i so (z^2-4z+5) is a factor.
z^4 – 6z^3 + 16z^2 - 22z + 15 = (z^2-4z+5)(z^2-2z+3)
use the quadratic formula to find the other two roots
√(3 - i) = 1.7553 - 0.2848i
√(3 - i) i = 0.2848 - 1.7553i
w = a * 1.7783 cis -0.1609
now finish it off. Not sure what the a has to do with anything. It's just a scalar multiplier.
Needs further calculation step by step.
c) Let z = √(3 - i)
i) Plot z on an Argand diagram:
To plot z = √(3 - i) on an Argand diagram, we need to find its real and imaginary parts.
We have z = √(3 - i) = √(3) * √(1 - (i/√3))
Since the argument of 1 - (i/√3) is -π/6, we have:
√(3) * √(1 - (i/√3)) = √3 * √(2e^(-iπ/6)) = √(6e^(-iπ/12))
Now, let's convert it into polar form:
r = √6
θ = -π/12
Plotting z on an Argand diagram, we draw a ray from the origin at an angle of -π/12, and mark the point where it intersects the circle with radius √6.
ii) Let w = az where a > 0, a ∈ R. Express w in polar form:
Let's express w = az in polar form.
We have w = az = √(6e^(-iπ/12))*a = a√6 * e^(-iπ/12)
The magnitude of w, r' = a√6, and the argument of w, θ' = -π/12.
iii) Express w^8 in the form ka^n(x + i√y) where k, x, y ∈ Z:
To express w^8 in the desired form, we need to calculate r'^8 and θ'*8.
r'^8 = (a√6)^8 = a^8 * 6^4 = 1296a^8
θ'*8 = -π/12 * 8 = -2π/3
Now, we have w^8 = 1296a^8 * e^(-2π/3)
Expressing in the desired form, ka^n(x + i√y):
w^8 = 1296a^8 * e^(-2π/3) = ka^n(x + i√y)
Where k = 1296a^8, n = 1, x = cos(2π/3), y = sin(2π/3).
To solve this problem, we will address each part step by step:
c) Let z = √(3 - i)
i) To plot z on an Argand diagram, we treat z as a complex number. The real part of z is the real part of the expression inside the square root, which is 3. The imaginary part of z is the imaginary part of the expression inside the square root, which is -1. Therefore, z is a complex number with a real part of 3 and an imaginary part of -1. On the Argand diagram, this corresponds to a point in the complex plane.
ii) Let w = az, where a > 0 and a ∈ ℝ. To express w in polar form, we need to find its magnitude (r) and argument (θ). The magnitude of a complex number z = x + yi is given by r = √(x^2 + y^2). In this case, we have z = √(3 - i) = 2 + i. Therefore, the magnitude of w is r = √[(a^2)(2^2 + 1^2)] = √(5a^2).
The argument of a complex number z = x + yi is given by θ = arctan(y / x). In this case, we have z = √(3 - i) = 2 + i. Therefore, the argument of w is θ = arctan(1/2).
Thus, w in polar form is expressed as w = r(cosθ + isinθ). Substituting the values we found earlier, we get w = √(5a^2)[cos(arctan(1/2)) + isin(arctan(1/2))].
iii) To express w^8 in the desired form ka^n(x + i√y), where k, x, and y are integers, we need to simplify the expression.
Using De Moivre's theorem, we have (cosθ + isinθ)^n = cos(nθ) + isin(nθ). In this case, we have w = √(5a^2)[cos(arctan(1/2)) + isin(arctan(1/2))].
To find w^8, we raise both the magnitude and the argument to the 8th power. The magnitude of w^8 is (r^8) = (5a^2)^4 = 625a^8. The argument of w^8 is 8θ = 8(arctan(1/2)).
Therefore, w^8 is expressed as w^8 = 625a^8[cos(8arctan(1/2)) + isin(8arctan(1/2))].
Overall, we have w^8 = 625a^8[cos(8arctan(1/2)) + isin(8arctan(1/2))], which is in the desired form ka^n(x + i√y) where k, x, and y are integers.