Chapter 4, Problem 63RE

To determine

The value of bounds to real zeroes of the following polynomial functions is to be estimated:

  $f\left(x\right)=x^3-x^2-4x+2$

Answer to Problem 63RE

The value of bounds to real zeroes of the given polynomial functions is $5$

Explanation of Solution

Given:

  $f\left(x\right)=x^3-x^2-4x+2$

Concept Used:

Bounds of the polynomial are expressed by the following equation:

  $Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\},1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}$

Calculation:

The given polynomial equation is:

  $f\left(x\right)=x^3-x^2-4x+2$

Get that:

Leading co-efficient: $1$

  $\begin{array}{l}{a}_0=2\\ a_1=-4\\ a_{n-1}=-1\end{array}$

Applying the bounds formula for the given polynomial as:

  $\begin{array}{l}Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,\left|2\right|+\left|-4\right|+\mathrm{...}+\left|-1\right|\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,2+4+1\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,7\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=7\end{array}$

And:

  $\begin{array}{l}1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+Max\left\{\left|2\right|,\left|-4\right|,\left|-1\right|\right\}\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+Max\left\{2,4,1\right\}\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+4\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=5\end{array}$

Hence, the bound of the polynomial is least numbers from two calculated values $5$ . Thus, every zero value of the given polynomial is occurred between interval $-5$ and $5$ .

So, the curve of the given polynomial is drawn as:

  

Conclusion:

Hence, the bound of zeroes of the given polynomial is $5$ .