Solution - Simplification or other simple results

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2".  4 more similar replacement(s).

Step  1  :

Equation at the end of step  1  :

  (((3•(x2))-5x)-(7•(x4)))-(((0-(2•(x3)))+(6•(x4)))-5x2)
 Step  2  :

Equation at the end of step  2  :

  (((3•(x2))-5x)-(7•(x4)))-(((0-(2•(x3)))+(2•3x4))-5x2)
 Step  3  :

Equation at the end of step  3  :

  (((3•(x2))-5x)-(7•(x4)))-(((0-2x3)+(2•3x4))-5x2)
 Step  4  :

Equation at the end of step  4  :

  (((3•(x2))-5x)-7x4)-(6x4-2x3-5x2)
 Step  5  :

Equation at the end of step  5  :

  ((3x2 -  5x) -  7x4) -  (6x4 - 2x3 - 5x2)

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -13x⁴ + 2x³ + 8x² - 5x  = 

  -x • (13x³ - 2x² - 8x + 5) 

Checking for a perfect cube :

 7.2    13x³ - 2x² - 8x + 5  is not a perfect cube

Trying to factor by pulling out :

 7.3      Factoring:  13x³ - 2x² - 8x + 5 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -8x + 5 
Group 2:  -2x² + 13x³ 

Pull out from each group separately :

Group 1:   (-8x + 5) • (1) = (8x - 5) • (-1)
Group 2:   (13x - 2) • (x²)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 7.4    Find roots (zeroes) of :       F(x) = 13x³ - 2x² - 8x + 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  13  and the Trailing Constant is  5.

 The factor(s) are:

of the Leading Coefficient :  1,13
 of the Trailing Constant :  1 ,5

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  -x • (13x3 - 2x2 - 8x + 5)