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 $\mathbf{\square }\mathbf{\left[}\mathbf{x}\mathbf{\right]}$.

$\left(c\right)\mathbf{}{\mathit{x}}^{\mathbf{3}}\mathbf{‐}{\mathit{x}}^{\mathbf{2}}\mathbf{‐}\mathbf{5}\mathit{x}\mathbf{+}\mathbf{5}$

Short answer

The factors as a product of irreducible polynomials are:

$$\begin{gathered} \square \left[x\right]=\left(x-1\right)\left(x^2-5\right) \\ \square \left[x\right]=\left(x-1\right)\left(x^2-5\right) \\ \square \left[x\right]=\left(x-1\right)\left(x+\sqrt{5}\right)\left(x-\sqrt{5}\right) \end{gathered}$$

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-x2-5x+5 .

The irreducible polynomial of$x^3-x^2-5x+5$are given below:

$$\begin{gathered} f\left(x\right)=x^3-x^2-5x+5 \\ \square \left[x\right]=\left(x-1\right)\left(x^2-5\right) \\ \square \left[x\right]=\left(x-1\right)\left(x^2-5\right) \\ \square \left[x\right]=\left(x-1\right)\left(x+\sqrt{5}\right)\left(x-\sqrt{5}\right) \end{gathered}$$

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

$$\begin{gathered} \square \left[x\right]=\left(x-1\right)\left(x^2-5\right) \\ \square \left[x\right]=\left(x-1\right)\left(x^2-5\right) \\ \square \left[x\right]=\left(x-1\right)\left(x+\sqrt{5}\right)\left(x-\sqrt{5}\right) \end{gathered}$$