Chapter 5.6, Problem 21PPS

To determine

To calculate: the remaining factors of the polynomial.

Answer to Problem 21PPS

The remaining factors are $\left(2x+7\right)\left(x+6\right)$

Explanation of Solution

Given information:

  $x-1$ is a factor of the polynomial $2x^3+17x^2+23x-42$

Formula used:

Synthetic Substitution :- Synthetic division is used to find the value of a function.

Factor Theorem :- The binomial $x-r$ is a factor of the polynomial $P(x)$ if and only if $P(r)=0$ .

Remainder Theorem:- If a polynomial $P(x)$ is divided by $x-r$ , the remainder s a constant $P(r)$ , and

  $P(x)=Q(x)(x-r)+P(r)$ ,

Where $Q(x)$ is a polynomial with degree one less than $P(x)$ .

Calculation:

Consider ,

  $2x^3+17x^2+23x-42$

Since, the binomial $x-1$ is a factor of the polynomial by using Synthetic division we get,

  $\begin{array}{l}\overline{)1}\begin{array}{cccc}2& 17& \text{ 23}& -42\end{array}\\ \begin{array}{ccc}\text{2}& \text{ 19}& \text{ 42}\end{array}\\ \overline{\begin{array}{ccc}2& \text{19}& \text{42}\end{array}\hspace{.2em}|}\begin{array}{c}\text{ 0}\end{array}\end{array}$

Thus, the remainder is zero. The polynomial $2x^3+17x^2+23x-42$ can be factored as $(x-1)(2x^2+19x+42)$ .

The polynomial $2x^2+19x+42$ is the depressed polynomial.

The remaining factors of the polynomial are :-

  $\begin{array}{l}=2x^2+19x+42\\ =2x^2+12x+7x+42\\ =2x(x+6)+7(x+6)\\ =(2x+7)(x+6)\end{array}$

Hence, the factors of the polynomial are $\left(x-1\right)\left(2x+7\right)\left(x+6\right)$ .