Chapter 5.6, Problem 1AGP

To determine

To calculate: the value of $f(3)$ .

Answer to Problem 1AGP

The value is 19.

Explanation of Solution

Given information:

  $f(x)=3x^3-6x^2+x-11$ .

Formula used:

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

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 ,

  $f(x)=3x^3-6x^2+x-11$

By using Remainder Theorem, $f(3)$ should be the remainder when the polynomial is divided by $x-3$ .

  $\begin{array}{l}\overline{)3}\begin{array}{cccc}3& -6& 1& -11\end{array}\\ \begin{array}{ccc}9& 9& 30\end{array}\\ \overline{\begin{array}{ccc}3& 3& 10\end{array}\hspace{.2em}|}\begin{array}{c}19\end{array}\end{array}$

The remainder is 19.

Therefore, by synthetic substitution, $f(3)=19.$