Question

(multiple choice) can anyone help me identify the real roots and state their multiplicity x^4+2x^23x^2-24x+144=0? ( :

Answer 1

root -4 multipicity 2, root 3multiplicity 2
root -4 multipicity 1, root 3multiplicity 2
root -4 multiplicity 2, root 3multiplicity 1
root -4 multipicity 1, root3 multiplicity 1

Answer 2

you gotta be kidding? a polynomial of degree 4??

Answer 3

i guess you are told that \(-4\) is a root so you can factor out the \(-4\) as \(x+4\) and see what is left
then try it again
me, i would cheat

Answer 4

I can't see the numbers you typed. It shows "{math processing Error}"

Answer 5

before we try it, there is a typo in your question
the thing you wrote above \(x^4+2x^3-2x^2-24x+144\) has no zeros

Answer 6

refresh your browser, it is not reading the latex correctly

Answer 7

check your question again
that sucker has no zeros

Answer 8

if I plug in 4 i get 464 as the sum. The zers is after the equal sign

Answer 9

I don't understand the multiplicity part :o

Answer 10

yes, there is a typo in the question

Answer 11

hm, It's straight from an assignment I'm working on. I'm not sure what the typo is

Answer 12

please check it again
the thing you wrote has no real zeros
once we get the right thing i can explain the "multiplicity" part, but not until we get the right expression to find the zeros of

Answer 13

if that is exactly what is written, then there is a mistake in the question so you cannot do it

Answer 14

woops I wrote 3-2x^2 it is 23x^2

Answer 15

x^4+2x^3-23x^2-24x+144=0

Answer 16

whew now we can do it!

Answer 17

haha I appreciate your patience ( :

Answer 18

actually we have to do very little
but before we do it, would you like the answer?

Answer 19

yes! :D

Answer 20

http://www.wolframalpha.com/input/?i=x^4%2B2x^3-23x^2-24x%2B144%3D0

Answer 21

you see where it says \((x-3)^2 (x+4)^2 = 0\) ?

Answer 22

yes those are the factors of 12

Answer 23

that tells you the zeros are \(3\) and \(-4\) and the fact that both of those factors has a exponent of \(2\) means the "multiplicity" of those zeros is 2

Answer 24

by "factor" i don't mean "factor of the numbers" i mean the factors are \(x-3\) and \(x+4\)
the exponent is the multiplicity
that shows up in the graph in the fact that the curve touches the \(x\) axis at those zeros but does not cross the axis

Answer 25

thank you : ) I send love into the universe for ya <3

Answer 26

lol
thanks
i didn't show you how to do it without cheating though
we can still do that if you like

Answer 27

yeah ( :

Answer 28

ok first off it is almost impossible to find the zeros of a 4th degree polynomial
but in this case you are given possible answers, and each answer has a 3 in it
so you can safely assume a zero is 3 and factor it out . i.e. factor \[x^4+2x^3-23x^2-24x+144\] as
\[(x-2)(something)\] and the easiest way to do that is by synthetic division

Answer 29

damn typo, i meant
\[x^4+2x^3-23x^2-24x+144\]\[(x-3)(something)\]

Answer 30

if you know how to do synthetic division this is quite easy
do you know it?

Answer 31

yes. bring down 1 * 3
3+2=5
5*3= 15
-23+15= -8
-8*3= -24
-24+-24 = -48??? I think I did something wrong

Answer 32

woops I forgot to multiply -48 by 3 and add it to 144 which leave a remainder of zero

Answer 33

yes you should have a remainder of zero because it factors, i.e. because if \(3\) is a zero, then you know \(x-3\) is a factor

Answer 34

what is your final answer?

Answer 35

(x-3)(x+4) x=3 x=-4?

Answer 36

oh no i mean after you factor
you get \[(x-3)(something)\] what was the "something"?

Answer 37

x+4?

Answer 38

after you did the synthetic division, what did you get?

Answer 39

0

Answer 40

2, 3, -5, 0

Answer 41

lol, not as the remainder! as the other part of the factor
it should have been \(x^3+5 x^2-8 x-48\)making
\[(x-3)(x^3+5 x^2-8 x-48)\]in factored form

Answer 42

in any case, you have to repeat this process with \(3\)
when you do it again, the last factor will be \(x^2+8x+16\) which is \((x+4)^2\) making the whole thing factor as \((x-3)^2(x+4)^2\) as we saw originally, and you are done

Answer 43

thank you ( : I understand it. I'm just confusing different problems :o

Answer 44

oh i see that you got it when you did the synthetic division
you got the coefficients as \(1,5,-8,-48\) making it what i wrote above
it is long and annoying, that is why i like to cheat

Answer 45

yw