Algebra Answers

Find the modulus and the principal argument of each of the given complex numbers.
(a) 3 − 4i,

(b) −2 + i,

(c) 1/1 + i √3

(d) (7 − i)/(-4 - 3i)

(e) 5(cos π/3 + isin π/3)

(f) cos 2π/3 − sin 2π/3.

Express the roots of the equation z3 − α3 = 0 in terms of α and w, where w is a complex cube root of unity. Use your answer to find the roots of the following equations in the form a + ib.

Express the roots of the equation z 3 − α 3 = 0 in terms of α and w, where w is a complex cube root of unity. Use your answer to find the roots of the following equations in the form a + ib. (a) z 3 − 27 = 0 (b) z 3 + 8 = 0 (c) z 3 − i = 0.

Determine the complex number z which satisfies the equations |z + 3i| = |z + 5 − 2i| and |z − 4i| = |z + 2i| simultaneously.

Show that z = i is a root of the equation z4 +z3 +z−1 = 0. Find the three other roots.

Given that 3 + i is a root of the equation z
3 3 3z
2 2 8z + 30 = 0, find the
remaining roots.

10. Given that w denotes either one of the non-real roots of the equation z
3 = 1, show
that
(a) 1 + w + w
2 = 0,

#117181
Express the roots of the equation z 3 − α 3 = 0 in terms of α and w, where w is a complex cube root of unity. Use your answer to find the roots of the following equations in the form a + ib. (a) z 3 − 27 = 0 (b) z 3 + 8 = 0 (c) z 3 − i = 0.

Given that w denotes either one of the non-real roots of the equation z3 = 1, show that (a) 1 +w+w2 =0, and 3 (b) the other non-real root is w2. Show that the non-real roots of the equation 1 −u u can be expressed in the form Aw and Bw2, where A and B are real numbers, find A and B.

Simplify the following expressions:
(a). (cos π/4 + isin π/4)(cos 3π/4 + isin 3π/4), (b). (cos π/4 + isin π/4)2
(cos π/6 + isin π/6) .