Chapter 6.6, Problem 2CYU

a.

To determine

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

Answer to Problem 2CYU

The value is $758$ .

Explanation of Solution

Given information:

  $f(x)=x^4+8x^3+x^2-4x-10$ .

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 ,

  $f(x)=x^4+8x^3+x^2-4x-10$

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

  $\begin{array}{l}\overline{)4}\begin{array}{cccc}1& 8& 1& \begin{array}{cc}-4& -10\end{array}\end{array}\\ \begin{array}{cccc}4& 48& 196& 768\end{array}\\ \overline{\begin{array}{cccc}1& 12& 49& 192\end{array}\hspace{.2em}|}\begin{array}{c}\text{ 758}\end{array}\end{array}$

The remainder is $758$ .

Therefore, by synthetic substitution, $f(4)=758.$

b.

To determine

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

Answer to Problem 2CYU

The value is $-46$ .

Explanation of Solution

Given information:

  $f(x)=x^4+8x^3+x^2-4x-10$ .

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 ,

  $f(x)=x^4+8x^3+x^2-4x-10$

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

  $\begin{array}{l}\overline{)-2}\begin{array}{cccc}1& 8& 1& \begin{array}{cc}-4& -10\end{array}\end{array}\\ \begin{array}{cccc}-2& -12& \text{ 22}& -36\end{array}\\ \overline{\begin{array}{cccc}1& 6& -\text{11}& \text{ 18}\end{array}\hspace{.2em}|}\begin{array}{c}-\text{46}\end{array}\end{array}$

The remainder is $-46$ .

Therefore, by synthetic substitution, $f(-2)=-46.$