Question

f(x)=x^2(x+2)(x-3)^4

Answer 1

What are you asked to do?

Answer 2

identify all of the zeros and state the multiplicity of eash

Answer 3

Oh. Okay :)
so, let's put it like this
\[\huge x^2(x+2)(x-3)^4\]

Answer 4

Or rather...
\[\huge (x-0)^2(x+2)(x-3)^4\]

Answer 5

Catch me so far?

Answer 6

A zero of a polynomial means at what value of x is the value of the function zero... :-)

Answer 7

yea.

Answer 8

Well, then, you see these parts?\[\huge (x\color{red}{-0})^2(x\color{red}{+2})(x\color{red}{-3})^4\]

If you take their respective negatives, then you get the zeros.
For instance, the negative of -0 is just 0. So 0 is a zero of the function.
What are the other zeros?

Answer 9

Sorry if "taking their respective negatives" sounds vague.
I actually mean multiply them by -1

Answer 10

so it would be -2 and 3

Answer 11

apart from 0, of course :)
Very good.
Now the multiplicity, to get them, I'll use 0 as an example again, the multiplicity of zero is dictated by this part (the exponent)
\[\huge (x\color{red}{-0})^{\color{blue}2}(x\color{red}{+2})(x\color{red}{-3})^4\]
that said, what are the multiplicities of -2 and 3?

Answer 12

im just a little confused

Answer 13

Multiplicity of 0 is 2, since the exponent of (x-0) is 2
So, what's the exponent of (x+2) ?
That's the multiplicity of -2

What's the exponent of (x-3) ?
That's the multiplicity of 3