Solution - Factoring binomials using the difference of squares

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "j2"   was replaced by   "j^2".  1 more similar replacement(s).

Step  1  :

            j3 + 5
 Simplify   ——————
            j2 + 2

Trying to factor as a Sum of Cubes :

 1.1      Factoring:  j³ + 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 :

 1.2    Find roots (zeroes) of :       F(j) = j³ + 5
Polynomial Roots Calculator is a set of methods aimed at finding values of  j  for which   F(j)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  j  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

Polynomial Roots Calculator :

 1.3    Find roots (zeroes) of :       F(j) = j² + 2

     See theory in step 1.2
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

Polynomial Long Division :

 1.4    Polynomial Long Division
Dividing :  j³ + 5 
                              ("Dividend")
By         :    j² + 2    ("Divisor")

Quotient :  j 
Remainder :  -2j + 5 

Trying to factor as a Sum of Cubes :

 1.5      Factoring:  j³ + 5 

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

Final result :

  j3 + 5
  ——————
  j2 + 2

See results of polynomial long division:

1. In step #01.04