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

`f(x) = x^5+7x^3-3x-12`

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Rational roots theorem

By the rational roots theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `-12` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational zeros are:

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

Note that when `abs(x) >= 3` we have:

`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 `x` with `abs(x) >= 3` can be a zero.

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 `f(x)` has no rational zeros, but has an irrational zero somewhere in `(1, 2)`

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Quintic

In fact this is a typical quintic with `1` Real zero and `4` non-Real complex zeros, none of which are even expressible in terms of radicals and elementary functions - including trigonometric, exponential or logarithmic ones.

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: