Chapter 2.4, Problem 47E

Solution

To check: All the zeros of the function $f\left(x\right)=x^5-4x^4-129x^3+396x^2+8x+3$ inside the window $\left[-1,4\right]$ .

Given information:

  $f\left(x\right)=x^5-4x^4-129x^3+396x^2+8x+3$

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

  $x=-1$ ,

  $\begin{array}{lllllll}-1\hfill & 1\hfill & -4\hfill & -129\hfill & 396\hfill & 8\hfill & 3\hfill \\ \hfill & \downarrow \hfill & -1\hfill & 5\hfill & 124\hfill & -520\hfill & 512\hfill \\ \hfill & 1\hfill & -5\hfill & -124\hfill & 520\hfill & -512\hfill & 515\hfill \end{array}$

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

  $x=4$ ,

  $\begin{array}{lllllll}4\hfill & 1\hfill & -4\hfill & -129\hfill & 396\hfill & 8\hfill & 3\hfill \\ \hfill & \downarrow \hfill & 4\hfill & 0\hfill & -516\hfill & -480\hfill & -1888\hfill \\ \hfill & 1\hfill & 0\hfill & -129\hfill & -120\hfill & -472\hfill & -1885\hfill \end{array}$

Since all the numbers in bottom row are not positive, the $x=4$ is not upper bound of the real roots.

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