Solution - Factoring binomials using the difference of squares

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   -x²⁷ - 8  =   -1 • (x²⁷ + 8) 

Trying to factor as a Sum of Cubes :

 2.2      Factoring:  x²⁷ + 8 

Theory : A sum of two perfect cubes,  a³ + b³ can be factored into  :
             (a+b) • (a²-ab+b²)
Proof  : (a+b) • (a²-ab+b²) =
    a³-a²b+ab²+ba²-b²a+b³ =
    a³+(a²b-ba²)+(ab²-b²a)+b³=
    a³+0+0+b³=
    a³+b³

Check :  8  is the cube of   2 
Check :  x²⁷ is the cube of   x⁹

Factorization is :
             (x⁹ + 2)  •  (x¹⁸ - 2x⁹ + 4) 

Trying to factor as a Sum of Cubes :

 2.3      Factoring:  x⁹ + 2 

Check :  2  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 2.4    Find roots (zeroes) of :       F(x) = x⁹ + 2
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  2.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor by splitting the middle term

 2.5     Factoring  x¹⁸ - 2x⁹ + 4 

The first term is,  x¹⁸  its coefficient is  1 .
The middle term is,  -2x⁹  its coefficient is  -2 .
The last term, "the constant", is  +4 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 4 = 4 

Step-2 : Find two factors of  4  whose sum equals the coefficient of the middle term, which is   -2 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  (-x9 - 2) • (x18 - 2x9 + 4)