Question 1049369
<pre>
{{{graph(400,400,-3.5,2.5,-200,200,8x^5+16x^4-20x^3+12)}}}

All the potential zeros are ± all fractions whose numerators
are divisors of the constant term, 12, and whose denominators 
are divisors of the leading coefficient 8.

The divisors of constant term 12 are (1,2,3,4,6,12}
The divisors of leading coefficient 8 are {1,2,4,8}

The only potential zeros are

{{{matrix(1,47,
"" +- 1/1,
",",
"" +- 1/2,
",", 
"" +- 1/4,
",", 
"" +- 1/8,
",",
"" +- 2/1,
",",
"" +- 2/2,
",", 
"" +- 2/4,
",", 
"" +- 2/8,
",",
"" +- 3/1,
",",
"" +- 3/2,
",", 
"" +- 3/4,
",", 
"" +- 3/8,
",",
"" +- 4/1,
",",
"" +- 4/2,
",", 
"" +- 4/4,
",", 
"" +- 4/8,
",",
"" +- 6/1,
",",
"" +- 6/2,
",", 
"" +- 6/4,
",", 
"" +- 6/8,
",",
"" +- 12/1,
",",
"" +- 12/2,
",", 
"" +- 12/4,
",", 
"" +- 12/8)}}}

and after reducing, we have:

{{{matrix(1,47,
"" +- 1,
",",
"" +- 1/2,
",", 
"" +- 1/4,
",", 
"" +- 1/8,
",",
"" +- 2,
",",
"" +- 1,
",", 
"" +- 1/2,
",", 
"" +- 1/4,
",",
"" +- 3,
",",
"" +- 3/2,
",", 
"" +- 3/4,
",", 
"" +- 3/8,
",",
"" +- 4,
",",
"" +- 2,
",", 
"" +- 1,
",", 
"" +- 1/2,
",",
"" +- 6,
",",
"" +- 3,
",", 
"" +- 3/2,
",", 
"" +- 3/4,
",",
"" +- 12,
",",
"" +- 6,
",", 
"" +- 3,
",", 
"" +- 3/2)}}}

And after eliminating the duplications, the
list shortens to:

{{{matrix(1,23,
"" +- 1,
",",
"" +- 1/2,
",", 
"" +- 1/4,
",", 
"" +- 1/8,
",",
"" +- 2,
",", 
"" +- 3,
",",
"" +- 3/2,
",", 
"" +- 3/4,
",", 
"" +- 3/8,
",",
"" +- 4,
",", 
"" +- 6,
",", 
"" +- 12)}}}

We see that the graph only crosses the x axis between -2 and -3,
and -3 is not a zero.  None of the others are between -2 and -3,
so it has no rational zeros, only the irrational one between -2 and
-3.  That means that the polynomial cannot be factored, and is 
therefore prime.

Edwin</pre>