Find and prove an upper bound on the number of times that two distinct polynomials of degree $d$ can intersect.
What if the polynomials' degrees differ?
My attempt:
let $p(x)$ and $q(x)$ be two distinct polynomials of degree $d$.
And they intersect if and only if $p(x_i)=q(x_i)$ for some $x_i$
Then the problem becomes to find the number of roots of $f(x) = p(x_i)-q(x_i) =0$
Since both of polynomials are degree of $d$, then $f(x)$ which is the difference between these two polynomials can only have degree of at most $d$, then the number of roots of $f(x)$ is $d$.
If the degrees of these two polynomial differ, then the number of roots of $f(x) =\max(\text{ degree of p}, \text{ degree of q}).$
I feel like I have done something wrong here since based on Paul's link, it should intersect at most $d^2$ times.