Chapter 2.4, Problem 46E

Solution

To check: All the zeros of the function $f\left(x\right)=x^5-x^4+21x^2+19x-3$ inside the window $\left[-3,2\right]$ .

Given information:

  $f\left(x\right)=x^5-x^4+21x^2+19x-3$

The graph is,

  

Concept Used:

Divide a polynomial function $g\left(x\right)$ by $\left(x-k\right)$ , where $k<0$ , using synthetic division and all numbers are alternative sign at last line, then $k$ is an lower bound to the real roots of the equation $g\left(x\right)=0$ .

Divide a polynomial function $g\left(x\right)$ by $\left(x-k\right)$ , where $k>0$ , using synthetic division and all positive numbers at last line, then $k$ is an upper bound to the real roots of the equation $g\left(x\right)=0$ .

Calculation:

Setup the synthetic division for $x=-3$ ,

  $\begin{array}{lllllll}-3\hfill & 1\hfill & -1\hfill & 0\hfill & 21\hfill & 19\hfill & -3\hfill \\ \hfill & \downarrow \hfill & -3\hfill & 12\hfill & -36\hfill & 45\hfill & -135\hfill \\ \hfill & 1\hfill & -4\hfill & 12\hfill & -15\hfill & 65\hfill & -138\hfill \end{array}$

Since the signs are alternative in bottom row, the $x=-3$ is lower bound of the real root.

Setup the synthetic division for $x=2$ ,

  $\begin{array}{lllllll}2\hfill & 1\hfill & -1\hfill & 0\hfill & 21\hfill & 19\hfill & -3\hfill \\ \hfill & \downarrow \hfill & 2\hfill & 2\hfill & 4\hfill & 50\hfill & 138\hfill \\ \hfill & 1\hfill & 1\hfill & 2\hfill & 25\hfill & 69\hfill & 135\hfill \end{array}$

Since all the numbers in bottom row are positive, the $x=2$ is upper bound of the real root.