Chapter 2.4, Problem 40E

Solution

To prove: $k=3$ is upper bound for the real zeros of the function $f\left(x\right)=4x^4-6x^3-7x^2+9x+2$ .

Given information:

  $f\left(x\right)=4x^4-6x^3-7x^2+9x+2$

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 $k=3$ ,

  $\begin{array}{cccccc}3& 4& -6& -7& 9& 2\\ & \downarrow & 12& 18& 33& 126\\ & 4& 6& 11& 42& 128\end{array}$

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

Hence proved.