Question

How do you find the zeros and use a sign chart to sketch the polynomial `F(x)=x^3(x+2)^2`?

Answer 1

Determination of the zeros and a sign chart of `f(x)` will only provide some limited information and not enough to sketch this equation. (see below)

Explanation:

Finding the zeros of `f(x)=x^3(x+2)^2`
Remember that a function has the value zero at (and only at) points where some factor of the function has the value zero.

The factors of `f(x)=x^2(x+2)^2`
are `x xx x xx x xx (x+2) xx (x+2)`
and the only unique factors are `x` and `(x+2)`

Therefore `f(x)` only has zeros where
#{:(x=0,color(white)("xxx")andcolor(white)("xxx"),(x+2)=0), (,,rarr x=-2):}#

For non-zero values of `f(x)` these points divide the domain into 3 intervals:
`x < -2color(white)("xxx") x in (-2,0)color(white)("xxx")x > 0`

We can pick arbitrary values within each interval to determine if `f(x)` is positive or negative in that interval.

#{: (x=-3,color(white)("xxx"),x=-1,color(white)("xxx"),x=1), (f(-3)=-27,,f(-1)=-1,,f(1)=+9), (f(x)" negative",,f(x)" negative",,f(x)" positive") :}#

Unfortunately this does not** tell us any detailed information about the behavior of `f(x)` except whether it is positive or negative within these ranges.

Just for reference, here is what the graph should look like, but you would need to use something beyond the zeros and a sign chart to sketch this.