Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  ((0-(6•(w4)))+(w2))-(((2•(w3))+22w2)-w)
 Step  2  :

Equation at the end of step  2  :

  ((0-(6•(w4)))+(w2))-((2w3+22w2)-w)
 Step  3  :

Equation at the end of step  3  :

  ((0 -  (2•3w4)) +  w2) -  (2w3 + 4w2 - w)

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   -6w⁴ - 2w³ - 3w² + w  = 

  -w • (6w³ + 2w² + 3w - 1) 

Checking for a perfect cube :

 5.2    6w³ + 2w² + 3w - 1  is not a perfect cube

Trying to factor by pulling out :

 5.3      Factoring:  6w³ + 2w² + 3w - 1 

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

Group 1:  3w - 1 
Group 2:  6w³ + 2w² 

Pull out from each group separately :

Group 1:   (3w - 1) • (1)
Group 2:   (3w + 1) • (2w²)

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 :

 5.4    Find roots (zeroes) of :       F(w) = 6w³ + 2w² + 3w - 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  w  for which   F(w)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  w  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  6  and the Trailing Constant is  -1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  -w • (6w3 + 2w2 + 3w - 1)