Question

How do you find all real zeros of the function `f(x)=2x(x-9)^2(x-2)^2`?

Answer 1

Real zeros at `x in {0,9,2}`

Explanation:

As an explicit expansion can be factored as:
`f(x)=color(red)(""(2x))xxcolor(blue)(""(x-9))xxcolor(green)(""(x-9))xxcolor(orange)(""(x-2))xxcolor(magenta)(""(x-2))`

Each factor gives one (not necessarily unique) zero:
`{: ("factor:",color(red)(""(2x)),color(blue)(""(x-9)),color(green)(""(x-9)),color(orange)(""(x-2)),color(magenta)(""(x-2))), ("implied zero:",color(white)("X")color(red)0,color(white)("X")color(blue)9,color(white)("X")color(green)9,color(white)("X")color(orange)2,color(white)("X")color(magenta)2) :}`