Question

How do you test for symmetry in an equation? For example, y = x to the fifth power + x to the third power + x?

Answer 1

To test for symmetry, first you need to know what kind of symmetry you want to test for. Is it Over-The-Origin Symmetry, or Over-The-X-Axis Symmetry, or Over-The-Y-Axis Symmetry?
First, you should rewrite your equation with new variables:
b = a^5 + a^3 + a
For ORIGIN SYMMETRY, substitute (-a,-b) for your x's and y's, respectively. Then solve your new equation to make it look like the equation above (so the b is not negative). Your solution should look like:
-b = (-a)^5 + (-3a)^3 + (-a)
b = -(-a^5) + -(-3a^3) + -(-a)
b = a^5 + 3a^3 + a
Because this equation is exactly the same with the equation above (b = a^5 + 3a^3 + a), it is symmetrical across the ORIGIN.
Now test it for X-axis and Y-axis using these variable substitutes.
X-Axis (a,-b)
Y-Axis (-a,b)
Make sure you double check your work!

Answer 2

Oops! I'm sorry for the typo, it should be (-a)^3 instead of (-3a)^3. All the typos after that are the same. My mistake, I'm sorry.