Describing a 2 error correcting BCH code by $c(x)$ codeword iff $β^1$, $β^2$, $β^3$, $β^4$ are roots of $c(x)$?

Question

I am trying to understand a statement I read about BCH codes. I don't get how we can determine $c(x)$ to be any codeword of BCH of length $2^r -1$ by just checking if $β^1$, $β^2$, $β^3$, $β^4$ are all roots of $c(x)$ P.S: Beta is a primitive element of Galois Field $GF(2^r)$ where r $>=$4 and $c(x)$ is in $K[x]$