complex algebra?

Question

(a)if w=4+3i and z= -3+2i,express
each of the following complex
numbers in cartesian form:-
(i)wz
(ii)conjugate of z
(iii)z^(-1)
(iv)w/z
(b)find a polynomial with real
coefficients whose roots are
(-2+7i) and (-2-7i)
(c)express the complex number
(-64) in polar form

note:- -64= -64+0i

Answer 1

Question a)
(i) wz = (4 + 3i) (-3 + 2i) = -12 - i - 6 = -18 - i

(ii) conj z = -3 - 2i

(iii) z^(-1) = 1/(-3 + 2i) = (-3 - 2i)/(-3 + 2i)(-3 - 2i)
= (-3 - 2i)/(9 + 4)
= (-1/13)(3 + 2i)

(iv) w/z = (4+3i)(-3 - 2i) / (-3 + 2i)(-3 - 2i)
= ( -12 -17i + 6) /13 = (- 6 -17i)/13 = -1/13(6 + 17i)

Question b)
Let z1 = - 2 + 7i and z2 = - 2 - 7i
(z - z1)(z - z2) = 0
z² - (z1 + z2) z + z1z2 = 0
z² - (-4)z + (-2 + 7i)(-2 -7i) = 0
z² + 4z + (4 + 49) = 0
z² + 4z + 53 = 0

Question c)
- 64 = 64(cos180° - i sin180°) = 64 at 180°

Answer 2

(4+3i)(-3+2i) = -12 +8i -9i +6i^2 = -18-i

cnj = -3-2i

1/z = 1/( -3+2i) = (-3-2i)/ 5 multiply num and den.by the conj of z

(4+3i)/(-3+2i) = (4+3i)( -3-2i) /5 = 1/5 ( (-6-17i)

(x-(-2-7i))( x- (-2+7i))= ((x+2)+7i)*((x+2) -7i) = (x+2)^2 +49=
x^2+4x+53

(-64) in polar has mod 64 and angl 180 deg.

Answer 3

i) (4 + 3i)(-3 + 2i) = -12 + 8i - 9i + 6i² = 12 - i - 6 = 6 - i

ii) -3 - 2i

iii) 1/(-3 + 2i) × (-3 - 2i)/(-3 - 2i) = (-3 - 2i)/(9 - 4i²) = (-3 - 2i)/(9 + 4) = (-3 - 2i)/13

iv) (4 + 3i)/(-3 + 2i) × (-3 - 2i)/(-3 - 2i) = (-12 - 8i - 9i - 6i²)/(9 - 4i²) = (-12 - 17i + 6)/(9 + 4) = (-6 - 17i)/13

b)

(x - (-2 + 7i))(x - (-2 - 7i)) = x² - (-2 - 7i)x - (-2 + 7i)x + (-2 + 7i)(-2 - 7i)) = x² + 2x + 7ix + 2x - 7ix + 4 + 14i - 14i - 49i² = x² + 4x + 4 + 49 = x² + 4x + 53

c) you sure you meant -64? In polar form that would just be r = 64, theta = π...did you forget the imaginary part?

Answer 4

i do no longer understand of a area, yet in a nutshell, any reactance (inductive or capacitive) could be represented by skill of a complicated impedance wherein the actual section is in many situations small. So, in case you are trying to study a community, graph it, label each and all of the aspects with their complicated impedance, and remedy the element utilising Kirchoff's rules and Thevenin and Norton equivalents, merely as you're able to if it have been all resistors. the only difference is which you're utilising complicated numbers rather of tangible ones.