Question

Find the polynomials $\mathit{q}\left(x\right)\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{r}\left(x\right)$such that $\mathit{f}\left(x\right)\mathbf{=}\mathit{g}\left(x\right)\mathit{q}\left(x\right)\mathbf{+}\mathit{r}\left(x\right)\mathbf{,}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{r}\left(x\right)\mathbf{}\mathit{o}\mathit{r}\mathbf{}\mathit{d}\mathit{e}\mathit{g}\mathbf{}\mathit{r}\left(x\right)\mathbf{<}\mathit{d}\mathit{e}\mathit{g}\mathbf{}\mathit{g}\left(x\right)\mathbf{:}$

$$\begin{gathered} \left(\mathit{a}\mathbf{\right)}\mathbf{}\mathit{f}\left(x\right)\mathbf{=}\mathbf{3}{\mathit{x}}^{\mathbf{4}}\mathbf{-}\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{6}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathit{x}\mathbf{+}\mathbf{2}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{g}\left(x\right)\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{}\mathit{i}\mathit{n}\mathbf{}\mathbf{\square }\left[x\right]\mathbf{.} \\ \mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{}\mathit{f}\left(x\right)\mathbf{=}{\mathit{x}}^{\mathbf{4}}\mathbf{-}\mathbf{7}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{g}\left(x\right)\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{}\mathit{i}\mathit{n}\mathbf{}\mathbf{\square }\left[x\right]\mathbf{.} \\ \mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{}\mathit{f}\left(x\right)\mathbf{=}\mathbf{}\mathbf{2}{\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{g}\left(x\right)\mathbf{=}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{}\mathit{i}\mathit{n}\mathbf{}{\mathbf{\square }}_{\mathbf{5}}\left[x\right]\mathbf{.} \\ \mathbf{\left(}\mathit{d}\mathbf{\right)}\mathbf{}\mathit{f}\left(x\right)\mathbf{=}\mathbf{4}{\mathit{x}}^{\mathbf{4}}\mathbf{+}\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{6}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{x}\mathbf{+}\mathbf{5}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{g}\left(x\right)\mathbf{=}\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{}\mathit{i}\mathit{n}\mathbf{}{\mathbf{\square }}_{\mathbf{7}}\left[x\right]\mathbf{.} \end{gathered}$$

Short answer

Answer:The required polynomials are:

$$\begin{gathered} \left(a\right)q\left(x\right)=3x^2-5x+8andr\left(x\right)=-4x-6in\square \left[x\right]. \\ \left(b\right)q\left(x\right)=\frac{1}{2}{x}^2-\frac{1}{4}andr\left(x\right)=-7x+\frac{5}{4}in\square \left[x\right]. \\ \left(c\right)q\left(x\right)=x^3+\frac{1}{2}{x}^2+\frac{3}{4}x-\frac{1}{8}andr\left(x\right)=\frac{7}{8}in{\square }_5\left[x\right]. \\ \left(d\right)q\left(x\right)=\frac{4}{3}{x}^2+\frac{2}{3}x+\frac{10}{9}andr\left(x\right)=\frac{8}{3}x+\frac{25}{9}in{\square }_7\left[x\right]. \end{gathered}$$

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.

Polynomials Division: (a)

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:

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

Hence, these are the required polynomials.

Polynomials Division: (b)

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:

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

Hence, these are the required polynomials.

Polynomials Division: (c)

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:

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

Hence, these are the required polynomials.

Polynomials Division: (d)

The given polynomials are:

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