The complex number z is such that |z| = 2 and arg(z) = \frac{\pi}{4} - HSC - SSCE Mathematics Extension 2 - Question 12 - 2015 - Paper 1

Now calculating u:

$u = z^2 = (\sqrt{2} + i\sqrt{2})^2 = 2 + 2\sqrt{2}i.$

To plot u on the Argand diagram, identify the point (2, 2\sqrt{2}). This point will also be in the first quadrant.

To find v, we first calculate \bar{z}:

$\bar{z} = \sqrt{2} - i\sqrt{2}.$

Now substituting:

$v = u - \bar{z} = (2 + 2\sqrt{2}i) - (\sqrt{2} - i\sqrt{2}) = 2 - \sqrt{2} + (2\sqrt{2} + \sqrt{2})i = 2 - \sqrt{2} + 3\sqrt{2}i.$

The point (2 - \sqrt{2}, 3\sqrt{2}) can now be plotted on the Argand diagram, which will also be in the first quadrant.

To find the roots, we will utilize synthetic division or factor theorem. After evaluating the polynomial, we can find:

$P(x) = (x - a - bi)(x - a + bi)(x - (a + 2bi))(x - (a - 2bi)).$

Let's determine a and b explicitly by substituting values into the polynomial and solving for them.

By using the previously found roots or performing polynomial long division, we can derive one quadratic polynomial:

$Q(x) = (x - (a + bi))(x - (a - bi)) = x^2 - 2ax + (a^2 + b^2).$

This will then present a simpler factorization of P(x).

The oblique asymptote will appear as y approaches infinity.

To sketch this graph:

1. Determine intercepts by setting y = 0 (which will occur at x = 2 and x = 5).
2. Identify the vertical asymptote at x = 1 where the denominator equals zero.
3. Indicate the oblique asymptote obtained from part (iii).
4. Draw the graph carefully to reflect these features.