Chapter 5.6, Problem 1CYU

a.

To determine

To calculate: the value of $f(4)$ .

Answer to Problem 1CYU

The value is $58$ .

Explanation of Solution

Given information:

  $f(x)=2x^3-5x^2-x+14$ .

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 ,

  $f(x)=2x^3-5x^2-x+14$

By using Remainder Theorem, $f(4)$ should be the remainder when the polynomial is divided by $x-4$ .

  $\begin{array}{l}\overline{)4}\begin{array}{cccc}2& -5& -1& 14\end{array}\\ \begin{array}{ccc}8& 12& 44\end{array}\\ \overline{\begin{array}{ccc}2& 3& 11\end{array}\hspace{.2em}|}\begin{array}{c}\text{ 58}\end{array}\end{array}$

The remainder is $58$ .

Therefore, by synthetic substitution, $f(4)=58.$

b.

To determine

To calculate: the value of $f(-2)$ .

Answer to Problem 1CYU

The value is $-20$ .

Explanation of Solution

Given information:

  $f(x)=2x^3-5x^2-x+14$ .

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 ,

  $f(x)=2x^3-5x^2-x+14$

By using Remainder Theorem, $f(-2)$ should be the remainder when the polynomial is divided by $x+2$ .

  $\begin{array}{l}\overline{)-2}\begin{array}{cccc}2& -5& -1& 14\end{array}\\ \begin{array}{ccc}-4& 18& -34\end{array}\\ \overline{\begin{array}{ccc}2& -9& 17\end{array}\hspace{.2em}|}\begin{array}{c}-\text{20}\end{array}\end{array}$

The remainder is $-20$ .

Therefore, by synthetic substitution, $f(-2)=-20.$