Abstract Algebra: Theory and Applications · Problem 17.5.1

The key is recognizing that $\mathbb{Z}_2[x]$ means polynomials with coefficients in $\{0, 1\}$ only—no other numbers exist in this field. A polynomial of degree 3 or less has the form $ax^3 + bx^2 + cx + d$ where each coefficient is independently either 0 or 1. Since you have 4 coefficient positions and 2 choices for each, you're really just listing all possible binary strings of length 4, then interpreting them as polynomials.

The key is recognizing that $\mathbb{Z}_2[x]$ means polynomials with coefficients in $\{0, 1\}$ only—no other numbers exist in this field. A polynomial of degree 3 or less has the form $ax^3 + bx^2 + cx + d$ where each coefficient is independently either 0 or 1. Since you have 4 coefficient positions and 2 choices for each, you're really just listing all possible binary strings of length 4, then interpreting them as polynomials.