Chapter 2.4, Problem 48E

Solution

To check: All the zeros of the function $f\left(x\right)=2x^5-5x^4-141x^3+216x^2-91x+25$ inside the window $\left[-5,5\right]$ .

Given information:

  $f\left(x\right)=2x^5-5x^4-141x^3+216x^2-91x+25$

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 $x=-5$ ,

  $\begin{array}{lllllll}-5\hfill & 2\hfill & -5\hfill & -141\hfill & 216\hfill & -91\hfill & 25\hfill \\ \hfill & \downarrow \hfill & -10\hfill & 75\hfill & 330\hfill & -2730\hfill & 14105\hfill \\ \hfill & 2\hfill & -15\hfill & -66\hfill & 546\hfill & -2821\hfill & 14130\hfill \end{array}$

Since the signs are not alternative in bottom row, the $x=-1$ is not lower bound of the real root.

Setup the synthetic division for $x=5$ ,

  $\begin{array}{lllllll}5\hfill & 2\hfill & -5\hfill & -141\hfill & 216\hfill & -91\hfill & 25\hfill \\ \hfill & \downarrow \hfill & 10\hfill & 25\hfill & -580\hfill & -1820\hfill & -9555\hfill \\ \hfill & 2\hfill & 5\hfill & -116\hfill & -364\hfill & -1911\hfill & -953\hfill \end{array}$

Since all the numbers in bottom row are not positive, the $x=5$ is not upper bound of the real roots.

The given window $\left[-5,5\right]$ is not correct for the given function.