The right answer is: The graph of the function is positive on (–6, –2)
So let's write in mathematical language each statement to solve this problem:
1. A polynomial function has a root of –6 with multiplicity 1
This can be written like this:
[tex] (x+6) [/tex]
2. It has a root of –2 with multiplicity 3
Writing this statement as follows:
[tex] (x+2)^3 [/tex]
3. It has a root of 0 with multiplicity 2
So, this statement is:
[tex] x^2 [/tex]
4. and it has a root of 4 with multiplicity 3
[tex] (x-4)^3 [/tex]
So, groping all these terms we have:
[tex] f(x)=x^2(x+6)(x+2)^3(x-4)^3 [/tex]
Moreover, the last statement says that the function has a positive leading coefficient and is of odd degree, say, this coefficient is 9, then the complete polynomial function is:
[tex] f(x)=9x^2(x+6)(x+2)^3(x-4)^3 [/tex]
From the Figure below you can see that the only right statement is that the graph of the function is positive on (–6, –2)
