Chapter 4, Problem 64RE

To determine

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

  $f\left(x\right)=x^3+x^2-10x-5$

Answer to Problem 64RE

The value of bounds to real zeroes of the given polynomial functions is $11$ .

Explanation of Solution

Given:

  $f\left(x\right)=x^3+x^2-10x-5$

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:

  $f\left(x\right)=x^3+x^2-10x-5$

Get that:

Leading co-efficient: $1$

  $\begin{array}{l}{a}_0=-5\\ a_1=-10\\ a_{n-1}=1\end{array}$

Applying the bounds formula for the given polynomial as:

  $\begin{array}{l}Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,\left|-5\right|+\left|-10\right|+\mathrm{...}+\left|-1\right|\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,5+10+1\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,16\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=16\end{array}$

And:

  $\begin{array}{l}1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+Max\left\{\left|-5\right|,\left|-10\right|,\left|-1\right|\right\}\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+Max\left\{5,10,1\right\}\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+10\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=11\end{array}$

Hence, the bound of the polynomial is least numbers from two calculated values $11$ . Thus, every zero value of the given polynomial is occurred between interval $-11$ and $11$ .

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

  

Conclusion:

Hence, the bound of zeroes of the given polynomial is $11$ .