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Example 3

Solve x9 - x5 + x4 - 1 = 0

Solution

x9 - x5+ x4 - 1 = x5 (x4 - 1) + 1(x4 - 1) = (x5+ 1) (x4 -1)

Now the 1st factor,

x5 + 1 = 0 Þ x5 = -1 = cos (2k + 1) p + i sin (2k + 1)p

Example 4

Solve x12 - 1 = 0 and which of its roots satisfy the equation x4 + x2 + 1 = 0

Solution

Therefore these roots satisfy both x12 - 1 = 0 and x4 + x2+ 1 = 0

Example 5

If a and b are roots of x2 - 2x + 4 = 0. Find an + bn. Hence a15 + b15

Solution

Example 6 Find all values of (1 + i)1/3and represent them on the Argand's diagram.

Solution

Thus z1 , z2 and z3 are the required all values of (l + i)1/3

They are represented on the Argand's diagram as

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Index

8.1 Geometry of complex numbers
8.2 De - Moivres's theorem
8.3 Roots of complex numbers
8.4 Cirsular functions of complex angles & hyperbolic function
Supplementary Problems

Chapter 9

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