Name: Basic Geometrical Concepts of Complex Number Plane
Uploaded: 2019-07-19
Duration: 5 min 49 s
Description: Before studying the geometrical applications of complex numbers, it is essential to recall the basic geometrical concepts of the complex number plane. Let's...

Question

Given that

z = 3 i

is a root, solve

z^{4} - 2 z^{3} + 14 z^{2} - 18 z + 45 = 0 .

Accepted Answer

Let $f (z) = z^{4} - 2 z^{3} + 14 z^{2} - 18 z + 45$

Given $z = 3 i$ is a root of $f (z)$ .

So, $z = - 3 i$ is also a root of $f (z)$ .

$\Rightarrow (z - 3 i) (z + 3 i) = z^{2} + 9$ is a factor of $f (z)$

Consider,

$\begin{array}{r} z^{2} - 2 z + 5 \\ z^{2} + 0 z + 9 z^{4} - 2 z^{3} + 14 z^{2} - 18 z + 45 \\ \frac{z^{4} + 0 z^{3} + 9 z^{2}}{- 2 z^{3} + 5 z^{2} - 18 z} \\ \frac{- 2 z^{3} + 0 z^{2} - 18 z}{5 z^{2} + 0 z + 45} \\ \frac{5 z^{2} + 0 z + 45}{0} \end{array}$

Thus, $z^{4} - 2 z^{3} + 14 z^{2} - 18 z + 45 = (z^{2} + 9) (z^{2} - 2 z + 5)$

Consider, $z^{4} - 2 z^{3} + 14 z^{2} - 18 z + 45 = 0$

$\Rightarrow (z^{2} + 9) (z^{2} - 2 z + 5) = 0$

$\Rightarrow (z^{2} + 9) = 0 or (z^{2} - 2 z + 5) = 0$

Now, $z = \pm 3 i$ is already given to be root.

Now, $z^{2} - 2 z + 5 = 0$

$\Rightarrow z^{2} - 2 z - 1 + 1 + 5 = 0$

$\Rightarrow (z - 1)^{2} - 1 + 5 = 0$

$\Rightarrow (z - 1)^{2} = - 4$

$\Rightarrow (z - 1) = \pm \sqrt{- 4}$

$\Rightarrow z - 1 = \pm 2 i [∵ i^{2} = - 1]$

$\Rightarrow z = 1 \pm 2 i$

Thus, the required solutions are $z = 3 i, - 3 i, 1 + 2 i, 1 - 2 i$ .

Given that $z = 3 i$ is a root, solve $z^{4} - 2 z^{3} + 14 z^{2} - 18 z + 45 = 0 .$

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Given that z=3i is a root, solve z4-2z3+14z2-18z+45=0.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Given that $z = 3 i$ is a root, solve $z^{4} - 2 z^{3} + 14 z^{2} - 18 z + 45 = 0 .$