Answer:
[tex]\newline{\text{f(x) has roots 0 multiplicity 3 and -2 multiplicity 2}}[/tex][tex]\bigskip\newline{\text{The root 0 has an odd multiplicity so it will cross the x-axis at the zero}}[/tex][tex]\bigskip\newline{\text{The root -2 has an even multiplicity so it willtouch the x-axis at the zero}}[/tex]
Step-by-step explanation:
[tex]\newline{\text{Assuming you mean }f(x)=-2x^3(x+2)^2 }[/tex][tex]\bigskip\newline{\text{If a function, f(x), can be factored into } f(x)=a(x-b)^c(x-d)^e\dots \text{ then, the function has roots b with multiplicity c and d with multiplicity e}}[/tex][tex]\bigskip\newline{\text{If the multiplicity is even, then the graph will touch the x-axis at the zero}}[/tex][tex]\bigskip\newline{\text{If the mutlpilcty is odd, then the graph will cross the x-axis at that zero}}[/tex][tex]\bigskip\newline{\text{Given } f(x)=-2x^3(x+2)^2}[/tex][tex]\bigskip\newline{\text{it can be factored into } f(x)=-2(x-0)^3(x-(-2))^2}[/tex][tex]\bigskip\newline{\text{Therefor f(x) has roots 0 multiplicity 3 and -2 multiplicity 2}}[/tex][tex]\bigskip\newline{\text{The root 0 has an odd multiplicity so it will cross the x-axis at the zero}}[/tex][tex]\bigskip\newline{\text{The root -2 has an even multiplicity so it willtouch the x-axis at the zero}}[/tex]
