Question

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

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

Short answer

Every zero of polynomial lies between $-5and+5$and the graph is

Step-by-step solution

Step 1. Given Information

The given polynomial is$f\left(x\right)=x^3-x^2-4x+2$

Step 2. Explanation

The leading coefficient of the polynomial is 1 with other coefficients as $a_2=-1,a_1=-4,a_0=2$

On applyiing the Max formula, we will find the bounds of the given polynomial.

role="math" localid="1646152870016" $$\begin{gathered} Max\left\{1,\left|a_0\right|+\left|a_1\right|+....+\left|a_{n-1}\right|\right\}=Max\left\{1,\left|2\right|+|-4|+|-1|\right\} \\ =Max\{1,2+4+1\} \\ =Max\{1,7\} \\ =7 \\ 1+Max\left\{\left|a_0\right|,\left|a_1\right|,....\left|a_{n-1}\right|\right\}=1+Max\left\{\right|2|,|-4|,|-1\left|\right\} \\ =1+Max\{2,4,1\} \\ =1+4 \\ =5 \end{gathered}$$

The smaller of the two numbers which means 5 is the bound of the polynomial.

So, every polynomial exist between$-5and+5$

Step 3. Graph

The graph of the given polynomial is