# Polynomials

• Mar 24th 2013, 08:59 PM
spotsymaj
Polynomials
Let a(x) and b(x) be in F[x]. I need to prove that if a(x) and b(x) determine the same function, and if the number of elements in F exceeds the degree of a(x) as well as the degree of b(x), then a(x) = b(x).

I know what if a(x) and b(x) determine the same function means. It means that a(x) - b(x) gives you the zero function. I dont know what if the number of elements in F exceeds the degree of a(x) as well as the degree of b(x) means. I also do not see how either of the two can conclude a(x) = b(x). Can anyone help me?
• Mar 26th 2013, 10:27 AM
Nehushtan
Re: Polynomials
The terminology is certainly confusing. I think what you mean by “zero function” is the constant function on $F$ whose image is 0, not the zero polynomial in $F[x]$. In other words, $a(x)$ and $b(x)$ determine the same function on F iff $a=b$ considered as functions $a,b:F\to F$.

An example to show what I mean. Suppose $F=\mathbb F_2=\{0,1\}$, $a(x)=x$, $b(x)=x^2$. Then, considered as functions $F\to F$, $a$ and $b$ are equal (to the identity function on $F$), but $a(x)\ne b(x)$ as polynomials in $F[x]$.

In the example, the number of elements of $F$ is not greater than the degree of $a(x)$. What you are asked to show is that if $|F|>\deg a,\deg b$, then $a(x)=b(x)$ as polynomials in $F[x]$ if $a=b$ as functions $F\to F$.