Question

Find a monic associate of

(a)$3x^3+2x^2+x+5in\mathrm{\mathbb{Q}}\left[x\right]$

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

(c)$i{x}^3+x-1in\mathrm{ℂ}\left[x\right]$

Short answer

Monic associates:

(a) $x^3+\frac{2}{3}{x}^2+\frac{x}{3}+\frac{5}{3}$

(b) $\left(x^5-\frac{4}{3}{x}^2+\frac{1}{3}\right)$

(c) $\left(x^3+\frac{x}{i}-\frac{1}{j}\right)$

Step-by-step solution

Monic associate

It is given that the polynomial is$3x^3+2x^2+x+5$ and we have to find the monic associate in $\mathrm{\mathbb{Q}}\left[x\right]$.

Steps for Monic associate

Take a common number from the polynomial.

(a).

$\begin{array}{rcl}p\left(x\right)& =& 3x^3+2x^2+x+5\\ & =& 3\left(x^3+\frac{2}{3}{x}^2+\frac{x}{3}+\frac{5}{3}\right)\end{array}$

Hence, $x^3+\frac{2}{3}{x}^2+\frac{x}{3}+\frac{5}{3}$ is a monic associate.

(b).

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

Hence, $\left(x^5-\frac{4}{3}{x}^2+\frac{1}{3}\right)$ is a monic associate.

(c).

$\begin{array}{rcl}p\left(x\right)& =& i{x}^3+x-1\\ & =& i\left(x^3+\frac{x}{i}-\frac{1}{i}\right)\end{array}$

Hence, $\left(x^3+\frac{x}{i}-\frac{1}{i}\right)$ is a monic associate.

Thus, the required answer is obtained.