Question

Find bounds to the real zeros of each polynomial function. Obtain a complete graph of f using a graphing utility.

$f\left(x\right)=2x^3-7x^2-10x+35$

Short answer

Every zero of polynomial lies between $-\frac{37}{2}and\frac{37}{2}$and the graph is

Step-by-step solution

Step 1. Given Information 

The given polynomial is$f\left(x\right)=2x^3-7x^2-10x+35$

Step 2. Explanation 

The leading coefficient of the polynomial is 2. So, first we will write the function such that the leading coefficient is 1.

$$\begin{gathered} f\left(x\right)=2(x^3-\frac{7}{2}{x}^2-\frac{10}{2}x+\frac{35}{2}) \\ =2(x^3-\frac{7}{2}{x}^2-5x+\frac{35}{2}) \end{gathered}$$

Now, leading coefficient is 1 with other coefficients as $a_2=-\frac{7}{2},a_1=-5,a_0=\frac{35}{2}$

On applyiing the Max formula, we will find the bounds of the given polynomial $$\begin{gathered} Max\left\{1,\left|a_0\right|+\left|a_1\right|+....+\left|a_{n-1}\right|\right\}=Max\left\{1,\left|\frac{35}{2}\right|+|-5|+|-\frac{7}{2}|\right\} \\ =Max\{1,\frac{35+10+7}{2}\} \\ =Max\{1,\frac{52}{2}\} \\ =\frac{52}{2} \\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,....\left|a_{n-1}\right|\right\}=1+Max\left\{\right|\frac{35}{2}|,|-5|,|\frac{7}{2}\left|\right\} \\ =1+Max\{\frac{35}{2},5,\frac{7}{2}\} \\ =1+\frac{35}{2} \\ =\frac{37}{2} \end{gathered}$$

The smaller of the two numbers which means $\frac{37}{2}$is the bound of the polynomial.

So, every polynomial exist between$-\frac{37}{2}and\frac{37}{2}$

Step 3. Graph

The graph of the given polynomial is