Chapter 2.4, Problem 43E

Solution

To prove: $k=0$ is upper bound for the real zeros of the function $f\left(x\right)=x^3-4x^2+7x-2$ .

Given information:

  $f\left(x\right)=x^3-4x^2+7x-2$

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 $g\left(x\right)=0$ .

Calculation:

Setup the synthetic division for $k=0$ ,

  $\begin{array}{l}\begin{array}{ccccc}0& 1& -4& 7& -2\\ & \downarrow & 0& 0& 0\\ & 1& -4& 7& -2\end{array}\\ \begin{array}{ccccc}& +& -& +& -\end{array}\end{array}$

Since the bottom rows numbers are alternative sign, the $k=0$ is lower bound.

Hence proved.