Solution - Factoring binomials using the difference of squares

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x5"   was replaced by   "x^5". 

Step  1  :

Equation at the end of step  1  :

  (0 -  (30 • (x2))) -  52x5
 Step  2  :

Equation at the end of step  2  :

  (0 -  (2•3•5x2)) -  52x5

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   -25x⁵ - 30x²  =   -5x² • (5x³ + 6) 

Trying to factor as a Sum of Cubes :

 4.2      Factoring:  5x³ + 6 

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 :

 4.3    Find roots (zeroes) of :       F(x) = 5x³ + 6
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  5  and the Trailing Constant is  6.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,2 ,3 ,6

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  -5x2 • (5x3 + 6)