Question

Perform the indicated operation and simplify your answer:

1. $\left(3x^4+2x^3-4x^2+x+4\right)+\left(4x^3+x^2+4x+3\right)in{\mathrm{\mathbb{Z}}}_5$
2. ${\left(x+1\right)}^{3}in{\mathrm{\mathbb{Z}}}_3\left[x\right]$
3. ${\left(x-1\right)}^{5}in{\mathrm{\mathbb{Z}}}_5\left[x\right]$
4. $\left(x^2-3x+2\right)\left(2x^3-4x+1\right)in{\mathrm{\mathbb{Z}}}_7$

Short answer

The required answers are:

1. $3x^4+x^3+2x^2+2$.
2. $x^3+1$.
3. $x^5-1$.
4. $2x^5+x^4+6x^2+3x+2$.

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.

Polynomial Operations:

(a)

The polynomial we have as:

$\left(3x^4+2x^3-4x^2+x+4\right)+\left(4x^3+x^2+4x+3\right)in{\mathrm{\mathbb{Z}}}_5$

Evaluating this, we have:

$\begin{array}{rcl}\left(3x^4+2x^3-4x^2+x+4\right)+\left(4x^3+x^2+4x+3\right)& =& 3x^4+6x^3-3x^2+5x+7\\ & =& 3x^4+x^3+2x^2+2\end{array}$

Hence, this is the required answer.

Polynomial Operations:

(b)

The polynomial we have:

${\left(x+1\right)}^{3}in{\mathrm{\mathbb{Z}}}_3\left[x\right]$

Evaluating this, we have:

$\begin{array}{rcl}{\left(x+1\right)}^3& =& x^3+3x^2+3x+1\\ & =& x^3+1\end{array}$

Hence, this is the required answer.

Polynomial Operations: 

(c)

The polynomial we have as:

${\left(x-1\right)}^{5}in{\mathrm{\mathbb{Z}}}_5\left[x\right]$

Evaluating this, we have:

$\begin{array}{rcl}{\left(x-1\right)}^5& =& x^5-5x^4+10x^3-10x^2+5x-1\\ & =& x^5-1\end{array}$

Hence, this is the required answer.

Polynomial Operations:

(d)

The polynomial we have as:

$\left(x^2-3x+2\right)\left(2x^3-4x+1\right)in{\mathrm{\mathbb{Z}}}_7$

Evaluating this, we have:

$\begin{array}{rcl}\left(x^2-3x+2\right)\left(2x^3-4x+1\right)& =& 2x^5-6x^4+13x^2-11x+2\\ & =& 2x^5+x^4+6x^2+3x+2\end{array}$

Hence, this is the required answer.