Chapter 2.4, Problem 39E

Solution

To prove: $k=2$ is upper bound for the real zeros of the function $f\left(x\right)=x^4-x^3+x^2+x-12$ .

Given information:

  $f\left(x\right)=x^4-x^3+x^2+x-12$

Concept Used:

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 $k=2$ ,

  $\begin{array}{cccccc}2& 1& -1& 1& 1& -12\\ & \downarrow & 2& 2& 6& 14\\ & 1& 1& 3& 7& 2\end{array}$

Since the bottom rows all numbers are positive, the $k=2$ is upper bound.

Hence proved.