Solution - Reducing fractions to their lowest terms

Step  1  :

            15
 Simplify   ——
            x2

Equation at the end of step  1  :

  15    
  —— +  3x
  x2    

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          3x     3x • x2
    3x =  ——  =  ———————
          1        x2   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 15 + 3x • x2     3x3 + 15
 ————————————  =  ————————
      x2             x2   

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   3x³ + 15  =   3 • (x³ + 5) 

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  x³ + 5 

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

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = x³ + 5
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  5.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,5

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  3 • (x3 + 5)
  ————————————
       x2