How do you find the zeroes for #f(x)=x^3(x-2)^2#?

1 Answer
Dec 9, 2015

Answer:

#x=0,2#

Explanation:

A zero is any point where #f(x)=0#.

So, we have to find when #x^3(x-2)^2=0#.

Notice how there are terms being multiplied by one another, equaling #0#. The only way things can have a product of #0# is if one of the things itself IS #0#.

So, to solve this, we can split apart #x^3(x-2)^2#.

We can say that any of these are true:

#x^3=0#

or

#(x-2)^2=0#

Solve for both of these:

#x^3=0#
#x=0#

#(x-2)^2=0#
#x-2=0#
#x=2#

Therefore #x=0,2# because both instances will give us an answer of #0#.

Look at a graph:
graph{x^3(x-2)^2 [-3.96, 7.14, -1.19, 4.357]}
The two zeros are indeed located at #0# and #2#.