Chapter 8.4, Problem 1OE

(a).

To determine

To express : the divisor as a polynomial for the given synthetic division.

Answer to Problem 1OE

The divisor of the given synthetic division is $x-2$

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)2}\text{ 1 }-3\text{ 5 }-4\\ \underset{\_}{\text{ 2 }-2\text{ 6}}\\ \text{ 1 }-1\text{ 3 | 2}\end{array}$

As per question,

In synthetic division, the divisor is of the form $x-p$ .

The synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{2}}}\stackrel{coefficientsofdividend}{\overbrace{\text{1 }-3\text{ 5 }-4}}\\ \underset{\_}{\text{ 2 }-2\text{ 6}}\\ \underset{coefficientsofquotients}{\underbrace{\text{1 }-1\text{ 3}}}\text{ |}\underset{remainder}{\underbrace{\text{2}}}\end{array}$

The divisor of the given synthetic division is $x-2$

(b).

To determine

To express : the dividend as a polynomial for the given synthetic division.

Answer to Problem 1OE

The dividend of the given synthetic division is $1x^3-3x^2+5x-4$

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)2}\text{ 1 }-3\text{ 5 }-4\\ \underset{\_}{\text{ 2 }-2\text{ 6}}\\ \text{ 1 }-1\text{ 3 | 2}\end{array}$

As per the question,

In synthetic division, the dividend is of the form $a{x}^3+b{x}^2+cx+d$ .

In the given synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{2}}}\stackrel{coefficientsofdividend}{\overbrace{\text{1 }-3\text{ 5 }-4}}\\ \underset{\_}{\text{ 2 }-2\text{ 6}}\\ \underset{coefficientsofquotients}{\underbrace{\text{1 }-1\text{ 3}}}\text{ |}\underset{remainder}{\underbrace{\text{2}}}\end{array}$

So, the dividend of the given synthetic division is $1x^3-3x^2+5x-4$ .

(c).

To determine

To express : the quotient as a polynomial for the given synthetic division.

Answer to Problem 1OE

The quotient of the given synthetic division is $1x^2-1x+3$ .

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)2}\text{ 1 }-3\text{ 5 }-4\\ \underset{\_}{\text{ 2 }-2\text{ 6}}\\ \text{ 1 }-1\text{ 3 | 2}\end{array}$

As per the question,

In synthetic division, the quotient is of the form $a{x}^2+bx+c$ .

The synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{2}}}\stackrel{coefficientsofdividend}{\overbrace{\text{1 }-3\text{ 5 }-4}}\\ \underset{\_}{\text{ 2 }-2\text{ 6}}\\ \underset{coefficientsofquotients}{\underbrace{\text{1 }-1\text{ 3}}}\text{ |}\underset{remainder}{\underbrace{\text{2}}}\end{array}$

So, the quotient of the given synthetic division is $1x^2-1x+3$ .

(d).

To determine

To express : the remainder as a polynomial for the given synthetic division.

Answer to Problem 1OE

The remainder of the given synthetic division is $2$ .

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)2}\text{ 1 }-3\text{ 5 }-4\\ \underset{\_}{\text{ 2 }-2\text{ 6}}\\ \text{ 1 }-1\text{ 3 | 2}\end{array}$

As per the question,

In synthetic division, the remainder is of the form $r$ .

The given synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{2}}}\stackrel{coefficientsofdividend}{\overbrace{\text{1 }-3\text{ 5 }-4}}\\ \underset{\_}{\text{ 2 }-2\text{ 6}}\\ \underset{coefficientsofquotients}{\underbrace{\text{1 }-1\text{ 3}}}\text{ |}\underset{remainder}{\underbrace{\text{2}}}\end{array}$

So, the remainder of the given synthetic division is $2$ .