Chapter 6.8, Problem 17PT

To determine

To find: the $f\left(-2\right)$ by using synthetic substitution method.

Answer to Problem 17PT

  $f\left(-2\right)=-33$ .

Explanation of Solution

Given:

  $f\left(x\right)=x^4-3x^3+5x-3$ .

Concept used:

In synthetic substitution method:

If $f\left(x\right)=a{x}^2+bx+c$ .

  $r\overline{)\begin{array}{l}\underset{\_}{\begin{array}{l}abc\\ ara{r}^2+br\end{array}}\\ aar+ba{r}^2+br+c\end{array}}$

Then the last remainder is the solution of the function.

  $f\left(r\right)=a{r}^2+br+c$ .

Calculation:

According to the given:

  $f\left(x\right)=x^4-3x^3+5x-3$ .

For $f\left(-2\right)$ :

  $-2\overline{)\begin{array}{l}\underset{\_}{\begin{array}{l}1\text{ -}35-3\\ 0-210\text{ -30 }\end{array}}\\ 1-515\text{ -33}\end{array}}$

There fore according to the synthetic substitution method the last remainder is the solution of the function.

  $f\left(-2\right)=-33$ .

It can be verified by putting the value of domain in the given equation.

Hence, $f\left(-2\right)=-33$ .