Question

In Problems 49– 60, for polynomial function

$f\left(x\right)=3\left(x^2+8\right){\left(x^2+9\right)}^2$

(a) List each real zero and its multiplicity.

(b) Determine whether the graph crosses or touches the x-axis at each x-intercept.

(c) Determine the maximum number of turning points on the graph.

(d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of .

Short answer

1. There are no real zero.
2. The graph of the function neither touches nor crosses x-axis at any point.
3. The turning points are 5.
4. The graph of function behave like graph of $3x^6$for large values of$\left|x\right|$.

Step-by-step solution

Step 1. Given Information       

The function is

$f\left(x\right)=3\left(x^2+8\right){\left(x^2+9\right)}^2$

Part (a) Step 1. To find zeros and their multiplicity.    

Equate given function with zero.

$$\begin{gathered} f\left(x\right)=0 \\ 3\left(x^2+8\right){\left(x^2+9\right)}^2=0 \end{gathered}$$

When

$$\begin{gathered} x^2+8=0 \\ {x}^2=-8 \end{gathered}$$

When

$$\begin{gathered} x^2+9=0 \\ {x}^2=-9 \end{gathered}$$

Since square of any real number can not be negative

So there is no real solution for these equation.

So no real zero of the function.

Part (b) Step 1. To determine whether the graph crosses or touches the x-axis at each x-intercept.     

No any x-intercepts for the function.

So the graph will not touch and cross the x-axis.

Part (C) Step 1. To determine the maximum number of turning points on the graph       

Expand the given function

$$\begin{gathered} f\left(x\right)=3\left(x^2+8\right){\left(x^2+9\right)}^2 \\ f\left(x\right)=3(x^2+8)(x^4+........) \\ f\left(x\right)=3(x^6+........) \end{gathered}$$

Here we can see that polynomial has degree 6

So turning point of the graph is $6-1=5$

Part (d). Step 1. To find end behavior of graph        

Since the highest power of $x$ after expanding the function is 6.

So the graph of the polynomial will behave like graph of$3x^6$

for large values of $\left|x\right|$..