## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.4 De Moivre's Theorem: Powers and Roots of Complex Numbers - 8.4 Exercises - Page 384: 52b

#### Answer

The value is approaching $w_1 = 1$. Therefore, the point $z_1 = 2+i$ should be colored red.

#### Work Step by Step

$f(z) = \frac{2z^3+1}{3z^2}$ $z_1 = 2+i$ We can evaluate the function for $z_1$: $z_2 = f(z_1)$ $z_2 = \frac{2(2+i)^3+1}{3(2+i)^2}$ $z_2 = \frac{2(2+11i)+1}{3(3+4i)}$ $z_2 = (\frac{5+22i}{9+12i})(\frac{9-12i}{9-12i})$ $z_2 = \frac{309+138i}{225}$ We can evaluate the function for $z_2$: $z_3 = f(z_2)$ $z_3 = \frac{2(\frac{309+138i~}{225}~)^3+1}{3(\frac{309+138i}{225})^2}$ $z_3 = \frac{2(\frac{11849841+36901062i}{11390625}~)+1}{3(\frac{76437+85284i}{50625})}$ $z_3 = \frac{(\frac{35090307+73802124i}{11390625})}{(\frac{229311+255852i}{50625})}$ $z_3 = [\frac{(\frac{35090307+73802124i}{225})}{229311+255852i}]~[\frac{229311-255852i}{229311-255852i}]$ $z_3 = \frac{(\frac{26929014418125+7945713630000i}{225})}{118043780625}$ $z_3 = 1.014+0.2992i$ We can see that the value is approaching $w_1 = 1$. Therefore, the point $z_1 = 2+i$ should be colored red.

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