Polynomial functions and their complex zeros are all about understanding where the function touches or crosses the x-axis.
Mechanistically, a zero is a solution to the equation p(x) = 0, where p(x) is your polynomial.
Complex zeros, like a + bi, show up in conjugate pairs if your polynomial has real coefficients.
This is because the coefficients' reality forces the imaginary parts to cancel out.
What trips you up is the multiplicity of these zeros.
If a zero has an even multiplicity, the graph just touches the x-axis and turns around.
If it's odd, the graph crosses through.
You might incorrectly assume every zero means a crossing, but that's not true for even multiplicities.
The degree of the polynomial, which you can figure out by looking at the highest power of x, tells you how many zeros you have, including these multiplicities.
Missing one zero or misunderstanding its nature means your graph won't match reality.