Question

Determine if the given polynomial is irreducible:

1. ${\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{7}\mathbf{\text{in}}\mathbf{ℝ}\left[x\right]$.
2. ${\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{7}\mathbf{\text{in}}\mathbf{\mathbb{Q}}\left[x\right]$.
3. ${\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{7}\mathbf{\text{in}}\mathbf{ℂ}\left[x\right]$.
4. $\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{+}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{2}\mathbf{\text{in}}{\mathbf{\mathbb{Z}}}_{\mathbf{5}}\left[x\right]$.
5. ${\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{9}\mathbf{\text{in}}{\mathbf{\mathbb{Z}}}_{\mathbf{11}}\left[x\right]$.
6. role="math" localid="1648621615519" ${\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{\text{in}}{\mathbf{\mathbb{Z}}}_{\mathbf{3}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$.

Short answer

1. It is proved that $x^2-7\text{in}\mathrm{ℝ}\left[x\right]$ is reducible.
2. It is proved that $x^2-7\text{in}\mathrm{\mathbb{Q}}\left[x\right]$ is irreducible.
3. It is proved that $x^2+7\text{in}\mathrm{ℂ}\left[x\right]$ is reducible.
4. It is proved that $2x^3+x^2+2x+2\text{in}{\mathrm{\mathbb{Z}}}_5\left[x\right]$ is irreducible.
5. It is proved that $x^3-9\text{in}{\mathrm{\mathbb{Z}}}_{11}\left[x\right]$is reducible.
6. It is proved that $x^4+x^2+1\text{in}{\mathrm{\mathbb{Z}}}_3\left[x\right]$ is reducible.

Step-by-step solution

Factor Theorem:

If F is any field such that $\mathit{f}\left(x\right)\mathbf{\in }\mathit{F}\left[x\right]$and $\mathit{k}\mathbf{\in }\mathit{F}$. Then, k will be the root of the polynomial $\mathit{f}\left(x\right)$, if and only if$\left(x-k\right)$ is a factor of $\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ in F[X].

Checking for Irreducible Polynomial:(a)

The given polynomial is: $x^2-7\text{in}\mathrm{ℝ}\left[x\right]$

Now, we have:

$x^2-7=\left(x-\sqrt{7}\right)\left(x+\sqrt{7}\right)$

Hence, $x^2-7\text{in}\mathrm{ℝ}\left[x\right]$ is reducible.

Checking for Irreducible Polynomial:(b)

The given polynomial is: $x^2-7\text{in}\mathrm{\mathbb{Q}}\left[x\right]$

Now, we have:

$x^2-7=\left(x-\sqrt{7}\right)\left(x+\sqrt{7}\right)$

Clearly, $\sqrt{7}\notin \mathrm{\mathbb{Q}}\left[x\right]$

Hence, $x^2-7\text{in}\mathrm{\mathbb{Q}}\left[x\right]$is irreducible.

Checking for Irreducible Polynomial:(c) 

The given polynomial is: $x^2+7\text{in}\mathrm{ℂ}\left[x\right]$

Now, we have:

$x^2+7=\left(x-i\sqrt{7}\right)\left(x+i\sqrt{7}\right)$

Hence, $x^2+7\text{in}\mathrm{ℂ}\left[x\right]$is reducible.

Checking for Irreducible Polynomial:(d)

The given polynomial is: $2x^3+x^2+2x+2\text{in}{\mathrm{\mathbb{Z}}}_5\left[x\right]$

Clearly, we have:

$f\left(b\right)\ne 0 ................\left\{\forall b\in {\mathrm{\mathbb{Z}}}_5\right\}$

Hence, $2x^3+x^2+2x+2\text{in}{\mathrm{\mathbb{Z}}}_5\left[x\right]$is irreducible.

Checking for Irreducible Polynomial:(e)

The given polynomial is:$x^3-9\text{in}{\mathrm{\mathbb{Z}}}_{11}\left[x\right]$

Now, we have:

$x^3-9=\left(x+7\right)\left(x^2+4x+5\right)$

Hence, $x^3-9\text{in}{\mathrm{\mathbb{Z}}}_{11}\left[x\right]$is reducible.

Checking for Irreducible Polynomial:(f)

The given polynomial is: $x^4+x^2+1\text{in}{\mathrm{\mathbb{Z}}}_3\left[x\right]$

Now, we have:

$x^4+x^2+1={\left(x+1\right)}^2{\left(x+2\right)}^2$

Hence, $x^4+x^2+1\text{in}{\mathrm{\mathbb{Z}}}_3\left[x\right]$is reducible.