Solution - Simplification or other simple results

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  :

  (3 • (x2)) -  24x5
 Step  2  :

Equation at the end of step  2  :

  3x2 -  24x5

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   3x² - 16x⁵  =   -x² • (16x³ - 3) 

Trying to factor as a Difference of Cubes:

 4.2      Factoring:  16x³ - 3 

Theory : A difference 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 :  16  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) = 16x³ - 3
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  16  and the Trailing Constant is  -3.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4 ,8 ,16
 of the Trailing Constant :  1 ,3

 Let us test ....

Note - For tidiness, printing of 15 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  -x2 • (16x3 - 3)