Question

1. Given the following polynomial, find
a. the zeros and the multiplicity of each
b. where the graph crosses or touches the x-axis
c. number of turning points
d. end behavior f(x)= (x-2)^3

Answer 1

Can you tell me where \[(x-2)^3 = 0\]?

Answer 2

What do you mean?

Answer 3

What values of \(x\) make \((x-2)^3 = 0\)?

Answer 4

What values of x? is the answer 0

Answer 5

It may be helpful to remember that \((x-2)^3 = (x-2)(x-2)(x-2)\)

\[(0-2)^3 = 0\]Really?

Answer 6

I have no clue... I suck when it comes to math, its been 20 years sence Ive seen or dealt with math....

Answer 7

But I am a quick learner, once you show me the formula, because I know its real simple, but knowing the formula, I can learn quick!!!

Answer 8

what is 0-2?

Answer 9

(2), so the value of x is 2

Answer 10

0-2 = -2, though maybe you're an accounting person, writing (2) instead?

yes, at \( x = 2, (x-2)^3 = 0\)

Answer 11

okay taking notes now.. no I'm not an accountant :), I don't know why I did that....

Answer 12

that's a typical financial report convention...but don't do it in math!

all right. as I hinted, our equation can be written as
\[f(x) = (x-2)^3 = (x-2)(x-2)(x-2) \]
and as you found, that equals 0 at \(x = 2\). So, our zero is at \(x = 2\).

Do you know or have a guess at what multiplicity means?

Answer 13

multiplicity of a member of a multiset is the number of times it appears in the multiset.

Answer 14

a large number or variety

Answer 15

okay. in this context, what do you think it means?

Answer 16

choice the largest number which is 2 (maybe)

Answer 17

hello you still there?

Answer 18

yes, I am. in this context, multiplicity means the number of times a root (or zero, as they are interchangeably called) is repeated. what is the multiplicity of x=2 in this equation?

Answer 19

3

Answer 20

very good.

okay, now we need to figure out where the graph crosses or touches the x-axis.

Answer 21

okay.....

Answer 22

some background: if the root has multiplicity 1, that's a spot where the graph crosses the x-axis. if the root has an odd multiplicity, that's a spot where the graph crosses the x-axis. if the root has an even multiplicity, that's a spot where it touches and moves away, like in my drawing:
(that's only supposed to be touching at a point, not a broad area)

Answer 23

for our function, we have an odd multiplicity. what does that mean?

Answer 24

1, or 3

Answer 25

no, what does it mean about the behavior of our graph

Answer 26

to be more precise, we have a zero with odd multiplicity...

Answer 27

that zero is at x = 2

what does the graph do at x=2?

Answer 28

go up 1 unit

Answer 29

Im guessing I don't know........

Answer 30

sigh. go back and read what I told you about multiplicity of roots

Answer 31

okay 3

Answer 32

I repeat:

some background: if the root has multiplicity 1, that's a spot where the graph crosses the x-axis. if the root has an odd multiplicity, that's a spot where the graph crosses the x-axis. if the root has an even multiplicity, that's a spot where it touches and moves away

Answer 33

from the multiplicity of a zero, you can determine the behavior of the graph at that value of x. at the value x = 2, where we have a zero of multiplicity 3, what will the graph do?

Answer 34

gose up 3 times

Answer 35

your only choices are "crosses the x-axis" and "touches the x-axis and retreats"

Answer 36

okay whpalmer, I do thank you for your help.. but I'm really a visual learner, If your trying to teach me i need someone in person that can show me.. I'm too confused, and frustrated.. Once again thanks for your help!!!!

Answer 37

Okay, do you know what a parabola looks like?

Answer 38

like the Mcdonalds arch