Chapter 8.4, Problem 5OE

(a).

To determine

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

Answer to Problem 5OE

The divisor of the given synthetic division is $x-3$

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)3}\text{ 1 }-3-1\text{ 5 }-\text{3}\\ \underset{\_}{\text{ 3 0 }-3\text{ 6}}\\ \text{ 1 }0-12\text{ | 3}\end{array}$

As per question,

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

The synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{3}}}\stackrel{coefficientsofdividend}{\overbrace{\text{1 }-3-15-\text{3}}}\\ \underset{\_}{\text{ 3 0 }-3\text{ 6}}\\ \underset{coefficientsofquotients}{\underbrace{\text{ 1 0 }-1\text{ 2}}}|\underset{remainder}{\underbrace{3}}\end{array}$

Here $p=3$

So, $x-p=x-3$

Therefore, the divisor of the given synthetic division is $x-3$

(b).

To determine

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

Answer to Problem 5OE

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

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)3}\text{ 1 }-3-1\text{ 5 }-\text{3}\\ \underset{\_}{\text{ 3 0 }-3\text{ 6}}\\ \text{ 1 }0-12\text{ | 3}\end{array}$

As per the question,

In the synthetic division, the dividend is a polynomial of the form $a{x}^4+b{x}^3+c{x}^2+dx+e$ .

In the given synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{3}}}\stackrel{coefficientsofdividend}{\overbrace{\text{1 }-3-15-\text{3}}}\\ \underset{\_}{\text{ 3 0 }-3\text{ 6}}\\ \underset{coefficientsofquotients}{\underbrace{\text{ 1 0 }-1\text{ 2}}}|\underset{remainder}{\underbrace{3}}\end{array}$

Here, $a=1,b=-3,c=-1,d=5$ and $e=-3$

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

(c).

To determine

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

Answer to Problem 5OE

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

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)3}\text{ 1 }-3-1\text{ 5 }-\text{3}\\ \underset{\_}{\text{ 3 0 }-3\text{ 6}}\\ \text{ 1 }0-12\text{ | 3}\end{array}$

As per the question,

In synthetic division, the quotient is a polynomial of the form $m{x}^3+n{x}^2+qx+s$ .

The synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{3}}}\stackrel{coefficientsofdividend}{\overbrace{\text{1 }-3-15-\text{3}}}\\ \underset{\_}{\text{ 3 0 }-3\text{ 6}}\\ \underset{coefficientsofquotients}{\underbrace{\text{ 1 0 }-1\text{ 2}}}|\underset{remainder}{\underbrace{3}}\end{array}$

Here, $m=1,n=0,q=-1$ , and $s=2$

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

(d).

To determine

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

Answer to Problem 5OE

The remainder of the given synthetic division is $3$

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)3}\text{ 1 }-3-1\text{ 5 }-\text{3}\\ \underset{\_}{\text{ 3 0 }-3\text{ 6}}\\ \text{ 1 }0-12\text{ | 3}\end{array}$

As per the question,

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

The given synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{3}}}\stackrel{coefficientsofdividend}{\overbrace{\text{1 }-3-15-\text{3}}}\\ \underset{\_}{\text{ 3 0 }-3\text{ 6}}\\ \underset{coefficientsofquotients}{\underbrace{\text{ 1 0 }-1\text{ 2}}}|\underset{remainder}{\underbrace{3}}\end{array}$

Here $r=3$

So, the remainder of the given synthetic division is $3$