A monic polynomial $p(x)$ of degree 4 has one repeated zero of multiplicity 2 and is divisible by $x^2 + x + 1$ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2020 - Paper 1

Since the polynomial is divisible by $x^2 + x + 1$, we first find the roots of this quadratic equation. The roots can be found using the quadratic formula:

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

where $a = 1$, $b = 1$, and $c = 1$. This results in complex roots, specifically:

$x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}$

The presence of complex roots implies that $p(x)$ will not cross the x-axis at these points.

Since one root is repeated at $x = r$, it will touch the x-axis at this point. The other complex roots imply that the polynomial will not cross the x-axis at those locations. Therefore, the graph must show the following characteristics:

1. Touching the x-axis at the repeated root
2. Not crossing at the complex roots, reflected in the overall shape of the graph.

Upon analyzing the characteristics of the polynomial, the graph option that shows a bounce at the x-axis corresponding to the repeated root while remaining above or below the axis at the complex roots is option C. This matches the required behavior of the polynomial, confirming that option C is the appropriate choice.