Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (((24•(x11))+(4•(x10)))-(6•(x9)))+2x8
 Step  2  :

Equation at the end of step  2  :

  (((24•(x11))+(4•(x10)))-(2•3x9))+2x8
 Step  3  :

Equation at the end of step  3  :

  (((24 • (x11)) +  22x10) -  (2•3x9)) +  2x8
 Step  4  :

Equation at the end of step  4  :

  (((23•3x11) +  22x10) -  (2•3x9)) +  2x8

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   24x¹¹ + 4x¹⁰ - 6x⁹ + 2x⁸  = 

  2x⁸ • (12x³ + 2x² - 3x + 1) 

Checking for a perfect cube :

 6.2    12x³ + 2x² - 3x + 1  is not a perfect cube

Trying to factor by pulling out :

 6.3      Factoring:  12x³ + 2x² - 3x + 1 

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

Group 1:  -3x + 1 
Group 2:  12x³ + 2x² 

Pull out from each group separately :

Group 1:   (-3x + 1) • (1) = (3x - 1) • (-1)
Group 2:   (6x + 1) • (2x²)

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 :

 6.4    Find roots (zeroes) of :       F(x) = 12x³ + 2x² - 3x + 1
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  12  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,3 ,4 ,6 ,12
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  2x8 • (12x3 + 2x2 - 3x + 1)