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

The polynomial must have 4 zeros in total. With one zero of multiplicity 2, we are left with 2 more zeros that can either be distinct or repeat a root that has been accounted for. The zeros associated with $x^2 + x + 1$ are complex and will not affect the degree.

Since the polynomial is of degree 4 (an even degree), its end behavior is that as $x$ approaches both positive and negative infinities, $p(x)$ approaches positive infinity.

Given the properties identified: the graph must touch the x-axis at the double root (indicating a local minimum or maximum) and will also feature the behavior corresponding to the complex roots. The graph must not cross the x-axis at the repeated zero.

By evaluating the characteristics of each graph option (A, B, C, D), option C exhibits a local minimum at the repeated zero while not crossing the x-axis, matching our described properties. Options A, B, and D do not satisfy the conditions for a polynomial with a repeated zero of multiplicity 2.