# Ask Uncle Colin: A difference of polynomials

Dear Uncle Colin,

Can two polynomials be equal everywhere in some non-trivial interval $[a,b]$ but not equal elsewhere?

- Lacking A Good Reasonable Answer, No Good Explanation

Hi, LAGRANGE, and thanks for your message!

The answer is “no”, but to explain why, I need to give you a few pieces of information first:

- The
*degree*of a polynomial is the highest power of $x$ anywhere in it - so $x^6 + 1$ is a degree-six polynomial. - A polynomial of degree $n$ or smaller has no more than $n$ zeroes (that is, there are at most $n$ places where $p(x) = 0$)
*unless*$p(x) = 0$ everywhere. - The difference between two polynomials, one of degree $m$ and one of degree $n$, is a polynomial degree of at most $\max(m,n)$.

So here’s the proof:

- Suppose $f(x)$ has degree $m$ and $g(x)$ has degree $n$ (without loss of generality, I’m going to assume $n>m$.)
- Consider $D(x) = f(x) - g(x)$. $D(x)$ is a polynomial of degree at most $n$.
- $D(x) = 0$ for all $x$ in the interval $[a,b]$.
- In particular, it is zero for
*at least*$n+1$ values of $x$ - Therefore $D(x) = 0$ everywhere.

- In particular, it is zero for
- Since $D(x) = 0$ everywhere, $f(x) = g(x)$ everywhere.

Hope that helps!

- Uncle Colin