Solution - Factoring binomials using the difference of squares

Step  1  :

               9  
 Simplify   ——————
            x3 + 9

Trying to factor as a Sum of Cubes :

 1.1      Factoring:  x³ + 9 

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

Polynomial Roots Calculator :

 1.2    Find roots (zeroes) of :       F(x) = x³ + 9
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  9.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 ,9

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

   d      9  
  —— • ——————
  dx   x3 + 9

Step  2  :

             d
 Simplify   ——
            dx

Canceling Out :

 2.1    Canceling out d as it appears on both sides of the fraction line

Equation at the end of step  2  :

  1      9  
  — • ——————
  x   x3 + 9

Step  3  :

Trying to factor as a Sum of Cubes :

 3.1      Factoring:  x³+9 

Check :  9  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  x³+9 

Check :  9  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Final result :

        9     
  ————————————
  x • (x3 + 9)