# Can #y=x^5 - x^3 + x - 2 x - 5# be factored? If so what are the factors ?

##### 1 Answer

Not algebraically.

#### Explanation:

I think there's at least one typo in the question, so consider three polynomials:

#f_1(x) = x^5-x^3+x-2x-5#

#f_2(x) = x^5-x^3+x^2-2x-5#

#f_3(x) = x^5+x^3+x^2+2x-5#

By the rational root theorem, the only rational zeros of these polynomials are expressible in the form

That means that the only possible rational zeros are:

#+-1# ,#+-5#

None of these are zeros of

Though they each have one Real zero and two pairs of Complex zeros there are no nice algebraic solutions to either.

In the case of

#x^5+x^3+x^2+2x-5 = (x-1)(x^4+x^3+2x^2+3x+5)#

The remaining quartic factor does have an algebraic solution, but it is not at all nice.