# How do you list all possible roots and find all factors of #x^5+7x^3-3x-12#?

##### 1 Answer

Possible *rational* zeros are:

#+-1, +-2, +-3, +-4, +-6, +-12#

but none are actually zeros.

This quintic is not solvable using radicals and elementary functions.

#### Explanation:

**Rational roots theorem**

By the rational roots theorem, any *rational* zeros of

That means that the only possible *rational* zeros are:

#+-1, +-2, +-3, +-4, +-6, +-12#

Note that when

#abs(7x^3)+abs(3x)+abs(12) <= abs(7x^3)+abs(3x)+abs(4x) = abs(7x^3)+abs(7x) < abs(7x^3)+abs(x^3) = abs(8x^3) < abs(x^5)#

So no

Checking the other possible rational zeros, we find:

#f(-2) = -32-56+6-12 = -94#

#f(-1) = -1-7+3-12 = -17#

#f(1) = 1+7-3-12 = -7#

#f(2) = 32+7-6-12 = 21#

So

**Quintic**

In fact this is a typical quintic with

About the best you can do is find approximations using numerical methods such as Durand Kerner.

See https://socratic.org/s/aAGsRKkf for another example and a description of the Durand-Kerner algorithm for a quintic.

Using this algorithm, I found the following approximations:

#x_1 ~~ 1.22622#

#x_(2,3) ~~ 0.101096+-2.734i#

#x_(4,5) ~~ -0.714207+-0.892944i#

Here's the C++ program I used: