Question

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept.
F(X)= 3(x-4)(x-1)^4

Answer 1

do you know how to find when the polynomial f is zero?

Answer 2

here is a hint if you have a*b=0 then a=0 or b=0 or both=0
so you solve x-4=0 and (x-1)^4=0
also you can determine the multiplicity of the zero by looking at the power

Answer 3

So you just plug 0 in where x is?

Answer 4

no

Answer 5

you plug in 0 where y is or where f(x) is..

Answer 6

But there is no Y in the equation?

Answer 7

y=f(x)
this is why I said or where f(x) is

Answer 8

To find the x-intercepts you set y to 0
To find the y-intercept you set x to 0
---
f(x)=y

Answer 9

\[y=3(x-4)(x-1)^4 \\ 0=3(x-4)(x-1)^4\]

Answer 10

so you then solve for x?

Answer 11

yes if you want to find the x-intercepts

Answer 12

I'm still confused on how to fine the answer for what I want to find???

Answer 13

did you read what I first said..
"here is a hint if you have a*b=0 then a=0 or b=0 or both=0
so you solve x-4=0 and (x-1)^4=0
also you can determine the multiplicity of the zero by looking at the power "

Answer 14

Yes, I read that...but (AGAIN) I am still confused...I don't understand what you mean??

Answer 15

so you are unable to solve x-4=0?
and also unable to solve (x-1)^4=0?

Answer 16

the first one would be x=4
I don't know how to do the second one...

Answer 17

do you know 0^4 is 0?

Answer 18

or you can think it like this
(x-1)^4=0
is equivalent to
(x-1)(x-1)(x-1)(x-1)=0

Answer 19

and again using that if you have a*b=0 then a=0 or b=0 or both=0

Answer 20

you have
x-1=0 or x-1=0 or x-1=0 or x-1=0

Answer 21

can you solve x-1=0?

Answer 22

x=1

Answer 23

right and you have that equation 4 times because of that power of 4
which tells you have x=1 multiplicity 4

Answer 24

ok, so does it touch or cross the x-axis?

Answer 25

you can look at a graph

but zeros associated with even multiplicities means you will have a touch
and
zeros associated with odd multiplicities means you will have a cross