Solutions to Polynomials of different degrees

Question

We have $p(x)$ a degree $m$ polynomial and $q(x)$ a degree $k$ polynomial. We also know that $p(x) = q(x)$ has at least $n+1$ solutions. And, $n\geq m\land n\geq k$.

Now, I tried graphing a little to see if I see a pattern

I tried making $$y= x^{2} $$ and $$y = -x^{2}+5 $$ There were two points of intersection so $n$ was $1$ in some sense here while the degrees were $2$.

Then the only way I thought that the condition can hold for $n+1$ solutions and $n \geq m $ and $ n \geq k$ is if $p(x)$ and $q(x)$ are the same polynomial.

That was the answer to this problem. But can someone give an idea for this please.

Answer 1

$P(x) - Q(x)$ is a polynomial of degree at most $\max(m,k)$, and therefore has at most $\max(m,k)$ roots unless it is the $0$ polynomial, i.e. unless $P = Q$.

Answer 2

just a hint

If the equation $R (x)=P (x)-Q (x)=0$ has at least $n+1$ solutions, it means that degree of $R (x)\ge n+1$.

but degree of $R (x)\le \max(m,k) $ thus $m\ge n+1$

or $k\ge n+1$