91影视

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.

The given polynomials are:

$$\begin{gathered} f\left(x\right)=3x^4-2x^3+6x^2-x+2 \\ g\left(x\right)=x^2+x+1 \end{gathered}$$

Now, dividing the polynomials as:

$\frac{3x^4-2x^3+6x^2-x+2}{x^2+x+1}=3x^2-5x+8withremainder-4x-6$

From this, we get:

$q\left(x\right)=3x^2-5x+8andr\left(x\right)=-4x-6in鈻�\left[x\right]$

Hence, these are the required polynomials.

The given polynomials are:

$$\begin{gathered} f\left(x\right)=x^4-7x+1 \\ g\left(x\right)=2x^2+1 \end{gathered}$$

Now, dividing the polynomials as:

$$\begin{gathered} \frac{x^4-7x+2}{2x^2+1}=\frac{1}{2}{x}^2-\frac{1}{4} \\ withremainder-7x+\frac{5}{4} \end{gathered}$$

From this, we get:

$q\left(x\right)=\frac{1}{2}{x}^2-\frac{1}{4}andr\left(x\right)=-7x+\frac{5}{4}in鈻�\left[x\right]$

Hence, these are the required polynomials.

The given polynomials are:

$$\begin{gathered} f\left(x\right)=2x^4+x^2-x+1 \\ g\left(x\right)=2x^{}-1 \\ Now,dividingthepolynomialsas: \\ \frac{2x^4+x^2-x+1}{2x-1}=x^3+\frac{1}{2}{x}^2+\frac{3}{4}-\frac{1}{8}withremainder\frac{7}{8} \end{gathered}$$

From this, we get:

$q\left(x\right)=x^3+\frac{1}{2}{x}^2+\frac{3}{4}x-\frac{1}{8}andr\left(x\right)=\frac{7}{8}in{鈻�}_5\left[x\right]$

Hence, these are the required polynomials.

The given polynomials are:

$$\begin{gathered} f\left(x\right)=4x^4+2x^3+6x^2+4x+5 \\ g\left(x\right)=3x^2+2 \\ Now,dividingthepolynomialsas: \\ \frac{4x^4+2x^3+6x^2+4x+5}{3x^2+2}=\frac{4}{3}{x}^2+\frac{2}{3}x+\frac{10}{9}withremainder\frac{8}{3}x+\frac{25}{9} \\ Fromthis,weget: \\ q\left(x\right)=\frac{4}{3}{x}^2+\frac{2}{3}x+\frac{10}{9}andr\left(x\right)=\frac{8}{3}x+\frac{25}{9}in{鈻�}_7\left[x\right] \\ Hence,thesearetherequiredpolynomials. \end{gathered}$$