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How to factor: x^5-6x^4+40x^2-13x-70 Please show me the steps!

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There is a theorem thing that states that plus or minus the factors of the constant(70 in this case) divided by plus or minus the constant in front of the term with the largest power are all the possibilities of the factors of the polynomial. So just start crunching the numbers.
try \[\pm1,2,5,7,10,35,70\]then use synthetic division or polynomial long division to divide out the factors
can you explain synthetic divsion please?

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let finish solving the problem and then I will explain
Then, can you please show the steps for doing the polynomial long division to divide out the factors
there is only one integer factor: (x-2). |dw:1357004129003:dw|
the other factor is (x-2)
What are the numbers of the thrid and forth row?
oh, I am beginning to see how synethetic divsion works
Please correct me if I am going to wrong way: x^5-6x^4+40x^2-13x-70 =(x-2)(x^4-4x^3-8x^2+24x+35) The next step would be to divide (x^4-4x^3-8x^2+24x+35) by 5 using synethetic divsion?
The remaining factors are not rational, so the rational roots theorem would not assist us any further.
oh, no wonder it didn't work anymore
so that's how you factor it
(x-2)(x^4-4x^3-8x^2+24x+35) This is the simplest form? Can't (x^4-4x^3-8x^2+24x+35) be factor too?
Since the remaining roots are complex, you couldn't factor out a single term, you would have to somehow factor out a quadratic. I do not know any techniques that will do this.
Not complex, irrational. But they exist as conjugates so you would have to factor out a quadratic instead of a single term.
ok, i will continue to try
I got it (x^4-4x^3-8x^2+24x+35)=(x^2-2x-7)(x^2-2x-5)
so when (x^5-6x^4+40x^2-13x-70) is factored, it will be (x-2)(x^2-2x-7)(x^2-2x-5) Is it correct?
Yes. That's it !
@yociyoci just curious, how did you factorize it?
Thank you so much everyone for helping me with factoring! I learnt so much from you guys!! Thanks!!
I plug in variables and did some algebra and trial and error
ya, it was long...
You can also use some facts as we usually do in quadratics. Assume (x^2+ax+b)(x^2+cx+d) then (a+c)=-4 ac+b+d=-8 bc+ad=24 bd=35 Hopefully this will help you eliminate cases quickly without having to multiply out the whole expression.
ya, I did something similar
That's a really good technique.
Great! Thank you for the enjoyable session!
Thank you very much, guys!! :)

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