One of the roots of the equation
$2z^4 - 14z^3 + 33z^2 - 26z + p = 0\text{, where } p \text{ is a constant}$
is $3 + \mathrm{i}$.

1. State the condition required for roots to occur in complex conjugate pairs and identify why it is not met here.

For a polynomial equation to necessarily have complex roots in conjugate pairs, all of its coefficients must be real numbers. The given information only states that $p$ is a constant, but does not specify that $p$ is a real number. Therefore, the coefficients may not all be real, and the complex roots might not occur in conjugate pairs.

1. Calculate the powers of $z = 3 + \mathrm{i}$.

$z = 3 + \mathrm{i}$
$z^2 = (3 + \mathrm{i})^2 = 9 + 6\mathrm{i} + \mathrm{i}^2 = 8 + 6\mathrm{i}$
$z^3 = z \cdot z^2 = (3 + \mathrm{i})(8 + 6\mathrm{i}) = 24 + 18\mathrm{i} + 8\mathrm{i} + 6\mathrm{i}^2 = 18 + 26\mathrm{i}$
$z^4 = z^2 \cdot z^2 = (8 + 6\mathrm{i})^2 = 64 + 96\mathrm{i} + 36\mathrm{i}^2 = 28 + 96\mathrm{i}$

2. Substitute $z$ and its powers into the polynomial equation.

$2(28 + 96\mathrm{i}) - 14(18 + 26\mathrm{i}) + 33(8 + 6\mathrm{i}) - 26(3 + \mathrm{i}) + p = 0$
$56 + 192\mathrm{i} - 252 - 364\mathrm{i} + 264 + 198\mathrm{i} - 78 - 26\mathrm{i} + p = 0$

3. Group real and imaginary parts to solve for $p$.

$(56 - 252 + 264 - 78 + p) + (192 - 364 + 198 - 26)\mathrm{i} = 0$
$(-10 + p) + 0\mathrm{i} = 0$
$p = 10$ (shown)

1. State the complex conjugate root due to the coefficients being real.

Since the coefficients of the polynomial equation $2z^4 - 14z^3 + 33z^2 - 26z + 10 = 0$ are all real numbers, the complex roots must occur in conjugate pairs.
Therefore, the complex conjugate of $3 + \mathrm{i}$, which is $z = 3 - \mathrm{i}$, is also a root.

2. Find the quadratic factor associated with these two known roots.

The quadratic factor corresponding to the roots $3 \pm \mathrm{i}$ is:
$(z - (3 + \mathrm{i}))(z - (3 - \mathrm{i})) = (z - 3)^2 - (\mathrm{i})^2$
$= (z^2 - 6z + 9) - (-1)$
$= z^2 - 6z + 10$

3. Determine the remaining quadratic factor by comparing coefficients.

Let the polynomial be expressed as the product of two quadratic factors:
$2z^4 - 14z^3 + 33z^2 - 26z + 10 = (z^2 - 6z + 10)(Az^2 + Bz + C)$
By comparing the $z^4$ coefficient: $A = 2$.
By comparing the constant term: $10C = 10 \implies C = 1$.
So the second factor is $(2z^2 + Bz + 1)$.
Comparing the $z^3$ coefficient:
$B - 6(2) = -14 \implies B - 12 = -14 \implies B = -2$
Thus, the second quadratic factor is $2z^2 - 2z + 1$.

4. Solve the second quadratic equation for the remaining roots.

Set the second factor to zero:
$2z^2 - 2z + 1 = 0$
Using the quadratic formula without a calculator:
$z = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(1)}}{2(2)}$
$z = \frac{2 \pm \sqrt{4 - 8}}{4}$
$z = \frac{2 \pm \sqrt{-4}}{4} = \frac{2 \pm 2\mathrm{i}}{4}$
$z = \frac{1}{2} \pm \frac{1}{2}\mathrm{i}$

5. Draw the Argand diagram plotting all four roots.

The four roots are $z_1 = 3 + \mathrm{i}$, $z_2 = 3 - \mathrm{i}$, $z_3 = \frac{1}{2} + \frac{1}{2}\mathrm{i}$, and $z_4 = \frac{1}{2} - \frac{1}{2}\mathrm{i}$.
On an Argand diagram, the horizontal axis is labelled $\text{Re}(z)$ and the vertical axis is labelled $\text{Im}(z)$.
The roots are plotted as points at coordinates $(3, 1)$, $(3, -1)$, $(0.5, 0.5)$, and $(0.5, -0.5)$.

1. State the coordinates of the roots on the Argand diagram and deduce the type of quadrilateral.

From part (c), the roots are $3 \pm \mathrm{i}$ and $\frac{1}{2} \pm \frac{1}{2}\mathrm{i}$.
The points on the Argand diagram are $(3, 1)$, $(3, -1)$, $(\frac{1}{2}, \frac{1}{2})$ and $(\frac{1}{2}, -\frac{1}{2})$.
Since the line segments connecting $(3, 1)$ to $(3, -1)$ and $(\frac{1}{2}, \frac{1}{2})$ to $(\frac{1}{2}, -\frac{1}{2})$ are both parallel to the imaginary axis, the quadrilateral has one pair of parallel sides.
Because the points are conjugate pairs, it is symmetrical about the real axis. Thus, it is an isosceles trapezium.

2. Calculate the area of the trapezium.

The lengths of the parallel sides are:
$1 - (-1) = 2$
$\frac{1}{2} - (-\frac{1}{2}) = 1$

The perpendicular distance (height) between the parallel sides is:
$3 - \frac{1}{2} = \frac{5}{2}$

$\text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$
$\text{Area} = \frac{1}{2} \times (2 + 1) \times \frac{5}{2} = \frac{15}{4} = 3.75$