Question

In Problems 49– 60, for polynomial function:$f\left(x\right)={\left(x-5\right)}^3{\left(x+4\right)}^2$

(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. The zeros of the function are 5 with multiplicity 3 and -4with multiplicity 2.
2. The graph of the function crosses x-axis at $x=5$and touches x-axis at $x=-4$.
3. The turning points are 4.
4. The graph of function behave like graph of $x^5$for large values of$\left|x\right|$

Step-by-step solution

Step 1. Given Information     

The function is $f\left(x\right)={\left(x-5\right)}^3{\left(x+4\right)}^2$

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

Equate given function with zero.

$$\begin{gathered} f\left(x\right)=0 \\ {\left(x-5\right)}^3{\left(x+4\right)}^2=0 \end{gathered}$$

When

$$\begin{gathered} {\left(x-5\right)}^3=0 \\ x-5=0 \\ x=5 \end{gathered}$$

When

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

So 5 is a zero with multiplicity 3 because power of $\left(x+5\right)$ is 3.

-4 is a zero of the function with multiplicity 2 because power of $\left(x+4\right)$is 2.

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

x-intercepts are 5 and -4

Since multiplicity of -4 is 2, which is even

So the graph of the function touches x-axis at this point.

Multiplicity of 5 is 3, which is odd

So graph of the function crosses x-axis at this point.

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

Expand the given function

$$\begin{gathered} f\left(x\right)={\left(x-5\right)}^3{\left(x+4\right)}^2 \\ f\left(x\right)=(x^3-5^3-3x^2.5+3.x.5^2)(x^2+8x+4^2) \\ f\left(x\right)=(x^3-15x^2+75x-125)(x^2+8x+16) \\ f\left(x\right)=x^5+8x^4+16x^3-15x^4-120x^3-240x^2+75x^3+600x^2+1200x-125x^2-1000x-2000 \end{gathered}$$

Here we can see that polynomial has degree 5

So turning point of the graph is $5-1=4$

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

Since the highest power of $x$ after expanding the function is 5.

So the graph of the polynomial will behave like graph of$x^5$

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