Question

1. List all polynomials of degree 3 in ${\mathrm{\mathbb{Z}}}_2\left[x\right]$.
2. List all polynomials of degree less than 3 in ${\mathrm{\mathbb{Z}}}_3\left[x\right]$.

Short answer

The required polynomials are:

1.

$$\begin{gathered} x^3 \\ {x}^3+1 \\ {x}^3+x \\ {x}^3+x^2 \\ {x}^3+x+1 \\ {x}^3+x^2+1 \\ {x}^3+x^2+x \\ {x}^3+x^2+x+1 \end{gathered}$$

2.

$$\begin{gathered} 0 \\ 1 \\ x \\ {x}^2 \\ x+1 \\ {x}^2+1 \\ {x}^2+x \\ {x}^2+x+1 \end{gathered}$$

Step-by-step solution

Polynomial Arithmetic:

If any given function $R\left[x\right]$ is a ring, then the commutative, associative, and distributive laws hold such that the function $f\left(x\right)+g\left(x\right)$ exists.

Polynomials: 

(a)

The polynomials of degree 3 in${\mathrm{\mathbb{Z}}}_2\left[x\right]$ can be given as:

$$\begin{gathered} x^3 \\ {x}^3+1 \\ {x}^3+x \\ {x}^3+x^2 \\ {x}^3+x+1 \\ {x}^3+x^2+1 \\ {x}^3+x^2+x \\ {x}^3+x^2+x+1 \end{gathered}$$

Hence, these are the required polynomials.

Polynomials: 

(b)

The polynomials of degree less than 3 in${\mathrm{\mathbb{Z}}}_3\left[x\right]$ can be given as:

$$\begin{gathered} 0 \\ 1 \\ x \\ {x}^2 \\ x+1 \\ {x}^2+1 \\ {x}^2+x \\ {x}^2+x+1 \end{gathered}$$

Hence, these are the required polynomials.