Question

what method can i use the factor this polynomial
X^4-12x^3+59x^2-138x+130
this one has no real zeros, the zeros are in complex system.

Answer 1

@zepdrix

Answer 2

I know the answer already, I want to know how to get it

Answer 3

Hmm not sure :C thinking....

Answer 4

take your time! I got this in exam. there was multiple choices i went ahead a looked at the ones that make sense and FOIL ed them, so I can get the original one haha. out of 4 choices a selected two that make sense. I got the right answer this way, but I'm unsatisfied haha

Answer 5

I was playing probability gamble lol

Answer 6

If you toss it into Wolfram, they factor it into,\[\Large\rm \left[(x-6)x+10\right]\left[(x-6)x+13\right]=0\]And then further into:\[\Large\rm \left[x^2-6x+10\right]\left[x^2-6x+13\right]=0\]And from there it's easy peasy finding the complex roots.
I'm not sure how they found the x-6 factors though.
wio or hartten might have some crazy method for doing this.

Answer 7

Oh! I was playing with something like this I did get x-6 thing in one side
but i was not sure where it was leading so I give up.
Thanks! let's ask these guys for more tips so I clear this for ever lol

Answer 8

@wio, @hartten

Answer 9

I will check what i did again and see

Answer 10

offline? darn :c

Answer 11

No worries, and thanks a lot.

Answer 12

@ganeshie8 some leads here!

Answer 13

:o I also want to know how to get (x-6).
Looks like a difficult but fun problem

Answer 14

I almost got something like that but time ran off it was an exam haha
I was like darn it, let's just use some other fake ways and get the result haha
So I foiled two options that make sense and get the answer, others were using TI to get that Factorization form zepdrix wrote

Answer 15

i remember i split 59 to 36 and 23 so i can complete the square with x^4-12x^3 part

Answer 16

i got x^2(x-6)^2

Answer 17

i need to do something similar with the other stuff haha, but time was going against me

Answer 18

needed*

Answer 19

@mathmale any idea to easily get this?

Answer 20

familiar with below precalc goodies ?
1) rational root theorem
2) descartes rule of signs
3) synthetic division
4) factor/reminder theorem

Answer 21

nvm, none of them gona work >.<

Answer 22

i'm familiar with those but they don't work here. if i could find one complex zero this solved done! but I could use any of those theorem. however I might be wrong, so i would like you to try it and see

Answer 23

@ikram002p what can say?

Answer 24

so basically you want some stable method to factor it into two quadratics ?

Answer 25

Correct!

Answer 26

since it has no real zeross , is it ok to give two quadratic combination ?

Answer 27

Yes! i needed a stable way to derive those quadratics

Answer 28

finding the quadratic factors is the tricky thing

Answer 29

mmmm ic :D
it might be more than one solution

Answer 30

it is! it involves may steps as i said i was working toward that but i give up haha

Answer 31

im thinking something like this may work :

\[x^4-12x^3+59x^2-138x+130 = (x^2+ax+b)(x^2+cx+d)\]

Answer 32

ikram you mean the zeros? they are 4!

Answer 33

but , tbh that wont be called factorize any more :o
its only give combinationns xD

Answer 34

compare the coefficients both sides and solve FOUR freaking equations !

Answer 35

then FOIL wow that might work ! rashhhhhhhh

Answer 36

actually u are right ikram the question asked linear combinations

Answer 37

rational, that sound good let me check it!

Answer 38

ikram i foiled hhh that's how i got the right answer i didn't have time for it hhh

Answer 39

oh you're talking about what ratio said lol

Answer 40

okay one second to see if that works

Answer 41

rashhh only Foil well not bad having 4 equation :3

Answer 42

wow so many equation haha

Answer 43

they don't look pleasant haha!
\(-12 = c+d\)
\(59 = d + ac + b\)
\(-138 = ad +bc\)
\(130 = bd\)

Answer 44

a+c=-12 no?

Answer 45

yes :) but lets give it up, they don't look solvable
\(-12 = c+a\)
\(59 = d + ac + b\)
\(-138 = ad +bc\)
\(130 = bd\)

Answer 46

it's too much haha

Answer 47

you give up, wth !
its only FOUR equations

Answer 48

the original question was only one equation :P

Answer 49

I don't like that equation haha

Answer 50

but the way sounded nice haha, thanks ratio

Answer 51

wolfram ?

Answer 52

okay how do we do it?
actually i have never heard of wolfram haha, may be i used but don't know what it is lol
my background is different

Answer 53

http://www.wolframalpha.com/input/?i=solve+-12+%3D+c%2Ba%2C+59+%3D+d+%2B+ac+%2B+b%2C+-138+%3D+ad+%2Bbc%2C+130+%3D+bd+over+reals

Answer 54

OMG!
hehe it did as what i thought xD
but i thought that was silly so didnt post hehe

Answer 55

that's cheating lol haha

Answer 56

what it did ?

Answer 57

i start from bd=130
and give possible values for b and d

Answer 58

wolfram fixed b

Answer 59

yeah that would be guess and check again, not systematic

Answer 60

hehe

Answer 61

I actually have no idea how that works?

Answer 62

mmm forget it lol
why it would be important to give combination anyway :P

Answer 63

haha, the question asked to do so, and i got 4 options

Answer 64

what are they ?

Answer 65

they are too much to write haha 4 linear combinations all of them
i foiled two of to get the answer easier than trying to get that first factorization haha

Answer 66

prof is expecting us to use the Texas Instrument

Answer 67

what are the zeros atleast?

Answer 68

:o Texas

Answer 69

hey @ikram002p actually your factoring is the best method here as 130 has very few factors and we can find the values of a,b,c,d very easily...

Answer 70

the zeros are x=3-i, 3+i, 3-2i, 3+2i
the question is saking factor as a product of linear factors

Answer 71

b and d can be one of : {2,5, 13}

Answer 72

ic:o
then it would be four possible equation ,wolfram gives 2

Answer 73

other students used TI while i was struggling haha

Answer 74

it should be 4 possible zeros since it is a 4th degree poly

Answer 75

idk what is texas or TI

Answer 76

4 possible linear combination ,this is what i ment

Answer 77

Graphing calculator

Answer 78

i don't have one hehe

Answer 79

ohh then wolfram is not cheating :P
since u suppose to solve it with calculator