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_{1} = - i

is a root of the equation

z^{3} + z^{2} + z + k = 0,

find the value of the constant

k .

Write down the other complex root of the equation and find the third root, stating whether it is real or complex.

Accepted Answer

Let $f (z) = z^{3} + z^{2} + z + k$

Given $z_{1} = - i$ is a root of $f (z)$ .

$\Rightarrow f (z_{1}) = 0$

$\Rightarrow {z_{1}}^{3} + {z_{1}}^{2} + z_{1} + k = 0$

$\Rightarrow {(- i)}^{3} + {(- i)}^{2} + (- i) + k = 0$

$\Rightarrow - i^{2} \cdot i + i^{2} - i + k = 0$

$\Rightarrow - (- 1) \cdot i - 1 - i + k = 0 [∵ i^{2} = - 1]$

$\Rightarrow i - 1 - i + k = 0$

$\Rightarrow k = 1$

Thus, the value of the constant $k = 1$ .

Thus, the equation becomes $z^{3} + z^{2} + z + 1 = 0$

We know that If $z_{1}$ is a root of the equation then $z_{1} * = z_{2}$ will also be a root, provided a polynomial has a degree at least two.

Now, $z_{2} = (- i) * = i$

So, $z - i$ and $z + i$ are factors of $f (z) = z^{3} + z^{2} + z + 1$

$\Rightarrow (z - i) (z + i)$ will also be factor of $f (z)$ .

Consider,

$\begin{array}{r} z + 1 \\ z^{2} + 0 z + 1 z^{3} + z^{2} + z + 1 \\ z^{3} + \underline{0 z^{2} + z} z^{2} + 1 \\ \frac{z^{2} + 1}{0} \end{array}$

i.e. $z^{3} + z^{2} + z + 1 = (z^{2} + 1) (z + 1)$

$\Rightarrow z = - 1$ is the third root and is real.

Given that $z_{1} = - i$ is a root of the equation $z^{3} + z^{2} + z + k = 0,$ find the value of the constant $k .$ Write down the other complex root of the equation and find the third root, stating whether it is real or complex.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Given that z1=-i is a root of the equation z3+z2+z+k=0, find the value of the constant k. Write down the other complex root of the equation and find the third root, stating whether it is real or complex.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Given that $z_{1} = - i$ is a root of the equation $z^{3} + z^{2} + z + k = 0,$ find the value of the constant $k .$ Write down the other complex root of the equation and find the third root, stating whether it is real or complex.