It is unusual to talk about symmetry about the x-axis, the y-axis, or the origin for a polar graph....
However, the problem can be answered.
To start with, the constants 5 and 3 have no effect on the answer to the problem; so we only need to consider the polar graph of cos(x).
For x-axis symmetry in a polar graph, it means we are replacing the angle x with the angle (360-x). So the question is whether cos(360-x) is equal to cos(x). The formula for the cosine of the difference of two angles tells us that it is:
So the graph of r = 5cos(3x) has symmetry about the x-axis.
(NOTE: You can get the same result by viewing x-axis symmetry as meaning the angle x gets replaced by the angle (-x).)
For y-axis symmetry in a polar graph, it means we are replacing the angle x with the angle (180-x). Is cos(180-x) equal to cos(x)? No:
The graph of r = 5cos(3x) does not have symmetry about the y-axis.
And for symmetry about the origin for a polar graph, it means we are replacing the angle x with the angle (x+180). Is cos(x+180) equal to cos(x)? No:
The graph of r = 5cos(3x) does not have symmetry about the origin.
ANSWER: The graph of r = 5cos(3x) has symmetry about the x-axis only.
