Chapter 8.4, Problem 3OE

(a).

To determine

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

Answer to Problem 3OE

The divisor of the given synthetic division is $x+1$

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)-1}\text{ 5 1 0 }6\text{ 2}\\ \underset{\_}{-5\text{ 4 }-4-2}\\ \text{ 5 }-4\text{ 4 2 | }0\end{array}$

As per question,

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

The synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{-1}}}\stackrel{coefficientsofdividend}{\overbrace{\text{5 }1\text{ 0 }6\text{ 2}}}\\ \underset{\_}{-5\text{ 4 }-4-2}\\ \underset{coefficientsofquotients}{\underbrace{\text{5 }-4\text{ 4 2}}}\text{ |}\underset{remainder}{\underbrace{0}}\end{array}$

Here $p=-1$

So, $x-p=x-(-1)=x+1$

Therefore, the divisor of the given synthetic division is $x+1$

(b).

To determine

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

Answer to Problem 3OE

The dividend of the given synthetic division is $5x^4+x^3+6x+2$

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)-1}\text{ 5 1 0 }6\text{ 2}\\ \underset{\_}{-5\text{ 4 }-4-2}\\ \text{ 5 }-4\text{ 4 2 | }0\end{array}$

As per the question,

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

In the given synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{-1}}}\stackrel{coefficientsofdividend}{\overbrace{\text{5 }1\text{ 0 }6\text{ 2}}}\\ \underset{\_}{-5\text{ 4 }-4-2}\\ \underset{coefficientsofquotients}{\underbrace{\text{5 }-4\text{ 4 2}}}\text{ |}\underset{remainder}{\underbrace{0}}\end{array}$

Here, $a=5,b=1,c=0,d=6$ and $e=2$

So, the dividend of the given synthetic division is $5x^4+x^3+6x+2$

(c).

To determine

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

Answer to Problem 3OE

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

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)-1}\text{ 5 1 0 }6\text{ 2}\\ \underset{\_}{-5\text{ 4 }-4-2}\\ \text{ 5 }-4\text{ 4 2 | }0\end{array}$

As per the question,

In the synthetic division, the quotient is of the form $s{x}^3+t{x}^2+ux+v$ .

The synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{-1}}}\stackrel{coefficientsofdividend}{\overbrace{\text{5 }1\text{ 0 }6\text{ 2}}}\\ \underset{\_}{-5\text{ 4 }-4-2}\\ \underset{coefficientsofquotients}{\underbrace{\text{5 }-4\text{ 4 2}}}\text{ |}\underset{remainder}{\underbrace{0}}\end{array}$

Here, $s=5,t=-4,u=4$ , and $v=2$

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

(d).

To determine

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

Answer to Problem 3OE

The remainder of the given synthetic division is $0$

Explanation of Solution

Given information :

The synthetic division:

  $\begin{array}{l}\overline{)-1}\text{ 5 1 0 }6\text{ 2}\\ \underset{\_}{-5\text{ 4 }-4-2}\\ \text{ 5 }-4\text{ 4 2 | }0\end{array}$

As per the question,

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

The given synthetic division:

  $\begin{array}{l}\overline{)\stackrel{p}{\overbrace{-1}}}\stackrel{coefficientsofdividend}{\overbrace{\text{5 }1\text{ 0 }6\text{ 2}}}\\ \underset{\_}{-5\text{ 4 }-4-2}\\ \underset{coefficientsofquotients}{\underbrace{\text{5 }-4\text{ 4 2}}}\text{ |}\underset{remainder}{\underbrace{0}}\end{array}$

Here $r=0$

So, the remainder of the given synthetic division is $0$