SOLUTION: Factor completely 2x^4 + x^3 - x^2 -x

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Question 367535: Factor completely 2x^4 + x^3 - x^2 -x
Answer by Jk22(389) About Me  (Show Source):
You can put this solution on YOUR website!
2x%5E4+%2B+x%5E3+-+x%5E2+-x

+x%282x%5E3+%2B+x%5E2+-+x+-+1%29


the polynomial inside is factorizable over R :

2x^3 +
2x+%28x%5E3+%2B+x%5E2%2F2+-+x%2F2+-+1%2F2%29

2x+%28+%28x+%2B+1%2F6%29%5E3+-+x%2F12+-+x%2F2+-+1%2F2+-+1%2F216%29

2x+%28+%28x+%2B+1%2F6%29%5E3+-+7x%2F12+-+109%2F216+%29+

let : x+%2B+1%2F6+=+y, we get in the parenthesis :

y%5E3+-+%287%2F12%29%2Ay+-+109%2F216+%2B+7%2F12+=+y%5E3+-+%287%2F12%29%2Ay+%2B+27%2F216


writing y+=+p+%2B+q,

p%5E3+%2B+q%5E3+%2B+3pq+%28p%2Bq%29+-+%287%2F12%29%2A%28p%2Bq%29+%2B+27%2F216+=+0

we get :

p%5E3+%2B+q%5E3+=+-27%2F216
3pq+=+7%2F12
the latter gives : q+=+7%2F36%2A1%2Fp, if substituted in the equation before :

p%5E3+%2B+%287%2F36%29%5E3%2Fp%5E3+=+-27%2F216 times p%5E3

p%5E6+%2B+%2827%2F216%29%2Ap%5E3+%2B+%287%2F36%29%5E3+=+0
with 2nd degree we find :
p%5E3+=+%28-27%2F216+%2B-+sqrt%28%2827%2F216%29%5E2+-+4%2A%287%2F36%29%5E3%29%29%2F2

q+=+7%2F36%2A1%2Fp and x++%2B+1%2F6+=+p+%2B+7%2F36%2A1%2Fp

with


hence the solution is

where 2 roots are identical.

we can write the polynomial as : 2x(x-x[++])(x-x[+-])(x-x[--])