Let $z_1 = 1 - 2i$, where $i^2 = -1$. (a) The complex number $z_1$ is a root of the equation $2z^3 - 7z^2 + 16z - 15 = 0$. Find the other two roots of the equation. (b) (i) Let $w = z_i ar{z_1}$, where $ar{z_1}$ is the conjugate of $z_1$. Plot $z$, $ar{z_1}$ and $w$ on the Argand diagram and label each point. (ii) Find the measure of the acute angle, $ar{z_1} w z_1$, formed by joining $ar{z_1}$ to $w$ to $z_1$ on the diagram above. Give your answer correct to the nearest degree.

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Let $z_1 = 1 - 2i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 2 - 2014

Question 2

Let $z_1 = 1 - 2i$, where $i^2 = -1$. (a) The complex number $z_1$ is a root of the equation $2z^3 - 7z^2 + 16z - 15 = 0$. Find the other two roots of the equation.... show full transcript

Worked Solution & Example Answer:Let $z_1 = 1 - 2i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 2 - 2014

Step 1

Find the other roots of the equation

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Answer

Given the polynomial equation is:

$2z^3 - 7z^2 + 16z - 15 = 0$

Since $z_1 = 1 - 2i$ is a root, we can use polynomial division to factor out $(z - (1 - 2i))$ from the equation. The result of the division gives us:

$(z - (1 - 2i))(2z^2 + z - 15)$

Next, we can find the roots of the quadratic equation $2z^2 + z - 15 = 0$ using the quadratic formula:

$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 2$ , $b = 1$ , and $c = -15$ .

Substituting these values, we calculate:

$z = \frac{-1 \pm \sqrt{1 + 120}}{4} = \frac{-1 \pm \sqrt{121}}{4} = \frac{-1 \pm 11}{4}$

Thus, we obtain:

$z = \frac{10}{4} = 2.5 \\ \text{and} \\ z = \frac{-12}{4} = -3$

Therefore, the other roots are $z_2 = 2.5$ and $z_3 = -3$ .

Step 2

Plot $z$, $ar{z_1}$ and $w$ on the Argand diagram

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Answer

In the Argand diagram:

Plot the point for $z = 1 - 2i$ , which is located at (1, -2).
The conjugate $\bar{z_1} = 1 + 2i$ is plotted at (1, 2).
For the point $w = \bar{z_1} z_1$ , first calculate: $\bar{z_1} z_1 = (1 + 2i)(1 - 2i) = 1^2 - (2i)^2 = 1 + 4 = 5$ Plot this point at (5, 0).

Label each point accordingly on the diagram.

Step 3

Find the measure of the acute angle, $ar{z_1} w z_1$

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Answer

To find the angle formed by the points $\bar{z_1}$ , $w$ , and $z_1$ , we can use the tangent function:

The lengths are:

$|\bar{z_1} w| = |(1 + 2i) - 5| = |(-4 + 2i)| = \sqrt{(-4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$
$|z_1 w| = |(1 - 2i) - 5| = |(-4 - 2i)| = 2\sqrt{5}$

Using the tangent formula:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{|y_2 - y_1|}{|x_2 - x_1|}$

Substituting the calculated values gives:

Opposite side: 2
Adjacent side: 4

Thus: $\tan^{-1}(\frac{2}{4}) = \tan^{-1}(\frac{1}{2}) \approx 26.57^{\circ}$

Final answer, rounding to the nearest degree:

$\theta \approx 53^{\circ}$

Let $z_1 = 1 - 2i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 2 - 2014

Worked Solution & Example Answer:Let $z_1 = 1 - 2i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 2 - 2014

Find the other roots of the equation

Plot $z$, $ar{z_1}$ and $w$ on the Argand diagram

Find the measure of the acute angle, $ar{z_1} w z_1$

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Plot $z$, $ar{z_1}$ and $w$ on the Argand diagram

Find the measure of the acute angle, $ar{z_1} w z_1$