Question

In Problems 49– 60, for polynomial function

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

(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 $\left|x\right|$.

Short answer

1. Zeros of the function are 0 with multiplicity 1 and $\pm \sqrt{3}$with multiplicity 1.
2. The graph of the function crosses x-axis at localid="1646306436707" $x=1,\pm \sqrt{3}$
3. The turning points are 2.
4. The graph of function behave like graph of $4x^3$for large values of $\left|x\right|$.

Step-by-step solution

Step 1. Given Information          

The function is

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

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

Equate given function with zero.

$$\begin{gathered} f\left(x\right)=0 \\ 4x(x^2-3)=0 \end{gathered}$$

when

$x=0$,

So this has multiplicity 1.

When

$$\begin{gathered} x^2-3=0 \\ {x}^2=3 \\ x=\pm \sqrt{3} \end{gathered}$$

it has multiplicity 1

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

x-intercepts for the function is 0 ,$\pm \sqrt{3}$with multiplicity 1 so graph of the function crosses x-axis.

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

Expand the given function

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

Here we can see that polynomial has degree 3

So turning point of the graph is$3-1=2$

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

Since the highest power of $x$after expanding the function is 3.

So the graph of the polynomial will behave like graph of$4x^3$

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