Chapter 4, Problem 66RE

To determine

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

  $f\left(x\right)=3x^3-7x^2-6x+14$

Answer to Problem 66RE

The value of bounds to real zeroes of the given polynomial functions is $\frac{17}{3}$ .

Explanation of Solution

Given:

  $f\left(x\right)=3x^3-7x^2-6x+14$

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:

  $\begin{array}{l}f\left(x\right)=3x^3-7x^2-6x+14\\ f\left(x\right)=3\left(x^3-\frac{7}{3}{x}^2-\frac{6}{3}x+\frac{14}{3}\right)\\ f\left(x\right)=3\left(x^3-\frac{7}{3}{x}^2-2x+\frac{14}{3}\right)\end{array}$

Get that:

Leading co-efficient: $1$

  $\begin{array}{l}{a}_0=\frac{14}{3}\\ a_1=-2\\ a_{n-1}=-\frac{7}{3}\end{array}$

Applying the bounds formula for the given polynomial as:

  $\begin{array}{l}Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,\left|\frac{14}{3}\right|+\left|-2\right|+\mathrm{...}+\left|-\frac{7}{3}\right|\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,\frac{14}{3}+2+\frac{7}{3}\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,\frac{27}{3}\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=Max\left\{1,9\right\}\\ Max\left\{1,\left|a_0\right|+\left|a_1\right|+\mathrm{...}+\left|a_{n-1}\right|\right\}=9\end{array}$

And:

  $\begin{array}{l}1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+Max\left\{\left|\frac{14}{3}\right|,\left|-2\right|,\left|-\frac{7}{3}\right|\right\}\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+Max\left\{\frac{14}{3},2,\frac{7}{3}\right\}\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=1+\frac{14}{3}\\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,\mathrm{...}\left|a_{n-1}\right|\right\}=\frac{17}{3}\end{array}$

Hence, the bound of the polynomial is least numbers from two calculated values; $\frac{17}{3}$ . Thus, every zero value of the given polynomial is occurred between interval $-\frac{17}{3}$ and $\frac{17}{3}$ .

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

  

Conclusion:

Hence, the bound of zeroes of the given polynomial is $\frac{17}{3}$ .