Question

Factor each polynomial as a product of irreducible polynomials in $\mathbf{\square }\left[x\right]$, in $\mathbf{\square }\mathbf{\left[}\mathbf{x}\mathbf{\right]}$, and in $\mathbf{\square }\mathbf{\left[}\mathbf{x}\mathbf{\right]}$.

$\left(b\right){\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1}$

Short answer

The factors as a product of irreducible polynomials are:

$$\begin{gathered} \square \left[x\right]=\left(x+1\right)\left(x^2‐x+1\right) \\ \square \left[x\right]=\left(x+1\right)\left(x^2‐x+1\right) \\ \square \left[x\right]=\left(x+1\right)\left(x‐\frac{1}{2}‐i\frac{\sqrt{3}}{2}\right)\left(x‐\frac{1}{2}+i\frac{\sqrt{3}}{2}\right) \end{gathered}$$

Step-by-step solution

Definition of Irreducible Polynomial.

A polynomial is called irreducible if it cannot be factorized into a non-trivial polynomial over the same field.

Finding the irreducible polynomial of x3+1 .

The given polynomial is$x^3+1$ .

The irreducible polynomial of$x^3+1$are given below:

$$\begin{gathered} f\left(x\right)=x^3+1 \\ \square \left[x\right]=\left(x+1\right)\left(x^2‐x+1\right) \\ \square \left[x\right]=\left(x+1\right)\left(x^2‐x+1\right) \\ \square \left[x\right]=\left(x+1\right)\left(x‐\frac{1}{2}‐i\frac{\sqrt{3}}{2}\right)\left(x‐\frac{1}{2}+i\frac{\sqrt{3}}{2}\right) \end{gathered}$$

Hence, the factors as a product of irreducible polynomials are:

$$\begin{gathered} \square \left[x\right]=\left(x+1\right)\left(x^2‐x+1\right) \\ \square \left[x\right]=\left(x+1\right)\left(x^2‐x+1\right) \\ \square \left[x\right]=\left(x+1\right)\left(x‐\frac{1}{2}‐i\frac{\sqrt{3}}{2}\right)\left(x‐\frac{1}{2}+i\frac{\sqrt{3}}{2}\right) \end{gathered}$$.