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

1. Identify the missing condition required for the conjugate root theorem.

The complex conjugate roots theorem states that if a complex number is a root of a polynomial, its complex conjugate is also a root, provided that all coefficients of the polynomial are real numbers.

In the given equation $2z^{4} - 14z^{3} + 33z^{2} - 26z + p = 0$, the constant $p$ is not explicitly stated to be a real number (it could be a complex number). Thus, the coefficients are not guaranteed to be entirely real, and the student's claim may be false.

1. Calculate the square, cube, and fourth power of $z = 3+i$.

Given $z = 3 + i$:

$z^2 = (3+i)^2 = 9 + 6i + i^2 = 8 + 6i$

$z^3 = z^2 \cdot z = (8+6i)(3+i)$
$z^3 = 24 + 8i + 18i + 6i^2 = 18 + 26i$

$z^4 = (z^2)^2 = (8+6i)^2$
$z^4 = 64 + 96i + 36i^2 = 28 + 96i$

2. Substitute $z$ and its powers into the equation and equate the parts to zero to find $p$.

Substitute the evaluated powers into the given equation:
$2z^4 - 14z^3 + 33z^2 - 26z + p = 0$

$2(28 + 96i) - 14(18 + 26i) + 33(8 + 6i) - 26(3 + i) + p = 0$

Expand and group the real and imaginary parts:
$(56 + 192i) - (252 + 364i) + (264 + 198i) - (78 + 26i) + p = 0$

$(56 - 252 + 264 - 78) + (192 - 364 + 198 - 26)i + p = 0$

$-10 + 0i + p = 0$

Solving for $p$ yields:
$p = 10 \quad \text{(shown)}$

1. Identify the second root using the property of real coefficients.

Since all coefficients of the equation $2z^4 - 14z^3 + 33z^2 - 26z + 10 = 0$ are real, the complex conjugate of $3+i$, which is $3-i$, must also be a root.

2. Form the quadratic factor corresponding to these two roots.

The quadratic factor is:
$[z - (3+i)][z - (3-i)]$
$= ((z-3) - i)((z-3) + i)$
$= (z-3)^2 - i^2$
$= z^2 - 6z + 9 - (-1)$
$= z^2 - 6z + 10$

3. Find the remaining quadratic factor by comparing coefficients.

Let the full polynomial be expressed as the product of two quadratics:
$2z^4 - 14z^3 + 33z^2 - 26z + 10 = (z^2 - 6z + 10)(2z^2 + bz + 1)$
Comparing the coefficients of $z^3$:
$b + 2(-6) = -14$
$b - 12 = -14 \implies b = -2$
So the remaining factor is $2z^2 - 2z + 1$.

4. Solve the remaining quadratic equation to find the other two roots.

Set the second factor to zero:
$2z^2 - 2z + 1 = 0$
Using the quadratic formula:
$z = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(1)}}{2(2)}$
$z = \frac{2 \pm \sqrt{4 - 8}}{4}$
$z = \frac{2 \pm \sqrt{-4}}{4}$
$z = \frac{2 \pm 2i}{4}$
$z = \frac{1}{2} \pm \frac{1}{2}i$

5. Mark the roots on an Argand diagram.

Draw and label the real axis ($\operatorname{Re}$) and imaginary axis ($\operatorname{Im}$). Plot and clearly label the four points:

• $(3, 1)$ for $z_1 = 3+i$
• $(3, -1)$ for $z_2 = 3-i$
• $(\frac{1}{2}, \frac{1}{2})$ for $z_3 = \frac{1}{2}+\frac{1}{2}i$
• $(\frac{1}{2}, -\frac{1}{2})$ for $z_4 = \frac{1}{2}-\frac{1}{2}i$

1. Determine the roots of the equation from part (c) to identify the vertices of the quadrilateral.

Since the coefficients are real, the complex conjugate $3-i$ is also a root.
The quadratic factor corresponding to these roots is $(z - (3+i))(z - (3-i)) = z^2 - 6z + 10$.
By long division or comparing coefficients, we factorise the quartic equation:
$2z^4 - 14z^3 + 33z^2 - 26z + 10 = (z^2 - 6z + 10)(2z^2 - 2z + 1) = 0$
Solving $2z^2 - 2z + 1 = 0$ gives $z = \frac{2 \pm \sqrt{4 - 8}}{4} = \frac{1}{2} \pm \frac{1}{2}i$.
The vertices of the quadrilateral on the Argand diagram are $(3, 1)$, $(3, -1)$, $(0.5, 0.5)$, and $(0.5, -0.5)$.

2. Identify the type of quadrilateral formed by the vertices.

The line segment connecting $(3, 1)$ and $(3, -1)$ is vertical.
The line segment connecting $(0.5, 0.5)$ and $(0.5, -0.5)$ is also vertical.
Since the shape has a pair of parallel sides and the vertices are symmetrically placed about the real axis, the quadrilateral is an isosceles trapezium.

3. Calculate the area of the trapezium.

The lengths of the parallel vertical sides are:
$1 - (-1) = 2$
$0.5 - (-0.5) = 1$

The perpendicular distance (horizontal height) between them is $3 - 0.5 = 2.5$.

$\text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$
$\text{Area} = \frac{1}{2}(2 + 1)(2.5) = 3.75$