Question

Find the bounds to the zeros of each polynomial function. Use the bounds to obtain a complete graph of f.

$f\left(x\right)=4x^4-12x^3+27x^2-54x+81$

Short answer

Every zero of the polynomial function will lie between$-21.25$ and $21.25$.

The graph of the function is

Step-by-step solution

Step 1. Given Information  

We are given a polynomial function $f\left(x\right)=4x^4-12x^3+27x^2-54x+81$.

We need to find the bounds to the zeros of the function and using it we need to obtain its graph.

Step 2. Concept used  

Let f denote a polynomial function whose leading coefficient is $1$.

$f\left(x\right)=x^n+a_{n-1}{x}^{n-1}+...+a_{1}x+a_0$

A bound $M$on the real zeros of f is the smaller of the two numbers

role="math" localid="1646153011423" $Max\left\{1,\left|a_0\right|+\left|a_1\right|+...+\left|a_{n-1}\right|\right\},1+Max\left\{\left|a_0\right|,\left|a_1\right|,...,\left|a_{n-1}\right|\right\}$

where Max $\left\{\right\}$means “choose the largest entry in $\left\{\right\}$.”

Step 3. Make the leading coefficient  1

The leading coefficient of the given polynomial is $4$. So divide the function side by$4$

$$\begin{gathered} f\left(x\right)=4x^4-12x^3+27x^2-54x+81 \\ f\left(x\right)=x^4-3x^3+\frac{27}{4}-\frac{27}{2}x+\frac{81}{4} \end{gathered}$$

On comparing with standard form we get

$$\begin{gathered} a_3=-3 \\ {a}_2=\frac{27}{4} \\ {a}_1=-\frac{27}{2} \\ {a}_0=\frac{81}{4} \end{gathered}$$

Step 4. Find the two numbers

Using the formula the two numbers are found as

$\begin{array}{rcl}Max\left\{1,\left|a_0\right|+\left|a_1\right|+\left|a_2\right|+\left|a_3\right|\right\}& =& Max\left\{1,\left|\frac{81}{4}\right|+\left|-\frac{27}{2}\right|+\left|\frac{27}{4}\right|+\left|-3\right|\right\}\\ & =& Max\left\{1,20.25+13.5+6.75+3\right\}\\ & =& Max\left\{1,43.5\right\}\\ & =& 43.5\end{array}$

and

$\begin{array}{rcl}1+Max\left\{\left|a_0\right|,\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}& =& 1+Max\left\{\left|\frac{81}{4}\right|,\left|-\frac{27}{2}\right|,\left|\frac{27}{4}\right|,\left|-3\right|\right\}\\ & =& 1+Max\left\{20.25,13.5,6.75,3\right\}\\ & =& 1+20.25\\ & =& 21.25\end{array}$

Step 5. Find the bounds 

Among the two numbers $43.5$and $21.25$, $21.25$is the smallest.

Therefore, $21.25$is the bound.

Every real zero off lies between $-21.25$and $21.25$.

Step 6. Graph the function 

Using the bound the graph of the function is given as