Find and prove an upper bound on the number of intersections on two distinct polynomials

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.

• Bézout's Theorem – Paul Dec 8 '11 at 7:57
• How many variables are the polynomials functions of? – Anthony Deluca Dec 8 '11 at 7:57
• Your approach is exactly the right approach. It yields a correct upper bound for the number of roots; examples are easily found to show that the upper bound is sharp. – Greg Martin Dec 8 '11 at 8:13
• only one variable, namely, $x$ if you will – geraldgreen Dec 8 '11 at 8:14
• @GregMartin but what about the link Paul posted? – geraldgreen Dec 8 '11 at 8:15

If the polynomials, $Q$ and $P$ have the same degree $d$, then the number of intersections points of their graphs is at most $d$, since these intersection points correspond to the zeroes of the polynomial $Q-P$. It is easy to see that this is a sharp estimate. For example, take $Q$ to be a polynomial with $d$ distinct real roots and set $P=-Q$.
If the degrees differ, the same argument shows that the number of intersection points of their graphs is at most $\max\{\text {deg}(P),\text{deg }(Q)\}$. This is a sharp estimate. For example take a polynomial with $r$ distinct real roots and write it as $$\underbrace{a_r x^r + a_{r-1} x^{r-1}+\cdots+a_{q+1} x^{q+1}}_Q+ \underbrace{a_q x^q+\cdots+a_1 x +a_0}_{-P}$$