Chapter 8.3, Problem 1E

a)

To determine

The remainder $r$ when $f\left(x\right)$ is divided by $x-c$ over the field $F$, where $f\left(x\right)=x^4-7x^2-3x+9$, $c=2$ and $F=R$.

b)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field $F$, where $f\left(x\right)=3x^5-2x^4+5x^2+2x-1$, $c=-1$, $F=R$.

c)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field $F$, where $f\left(x\right)=x^4-i{x}^3+3x^2-3ix$, $c=i$ and $F=C$.

d)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field $F$, where $f\left(x\right)=x^4-i{x}^3+3x^2-3ix$, $c=-i$ and $F=C$.

e)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field $F$, $f\left(x\right)=x^5+x^3+x+1$, $c=1$ and $F=Z_3$.

f)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field $F$, where $f\left(x\right)=x^4+x^3+2x^2+1$, $c=2$ and $F={\text{Z}}_3$.

g)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field $F$, where $f\left(x\right)=x^3+4x^2+2x+1$, $c=3$ and $F=Z_5$.

h)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field $F$, where $f\left(x\right)=2x^4+3x^3+4x^2+3$, $c=2$ and $F=Z_5$.

i)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field $F$, where $F=Z_7$, $f\left(x\right)=x^4+5x^3+2x^2+6x+2$, $c=4$.

j)

To determine

The remainder of the polynomial $f\left(x\right)$ when divided by $x-c$ over the field F, where $f\left(x\right)=x^3+6x^2+2x+2$, $c=5$ and $F=Z_7$.