Chapter 6.6, Problem 4CYU

To determine

To calculate: the remaining factors of the polynomial.

Answer to Problem 4CYU

The remaining factors are $(x-2)(x-3)$

Explanation of Solution

Given information:

  $x-1$ is a factor of the polynomial $x^3-6x^2+11x-6$

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 ,

  $x^3-6x^2+11x-6$

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

  $\begin{array}{l}\overline{)1}\begin{array}{cccc}1& -6& 11& -6\end{array}\\ \begin{array}{ccc}1& -5& 6\end{array}\\ \overline{\begin{array}{ccc}1& -5& 6\end{array}\hspace{.2em}|}\begin{array}{c}\text{ 0}\end{array}\end{array}$

Thus, the remainder is zero. This implies polynomial $x^3-6x^2+11x-6$ can be factored as $(x-1)(x^2-5x+6)$ .

The polynomial $x^2-5x+6$ is the depressed polynomial.

The remaining factors of the polynomial are :-

  $\begin{array}{l}=x^2-5x+6\\ =x^2-2x-3x+6\\ =x(x-2)-3(x-2)\\ =(x-2)(x-3)\end{array}$

Hence, the factors of the polynomial are $(x-1)(x-2)(x-3)$ .