Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "/-3x" was replaced by "/(-3x)".

Step  1  :

Equation at the end of step  1  :

  ((((6•(x3))+11x2)-9x)+12)
  —————————————————————————-1
             -3x           
 Step  2  :

Equation at the end of step  2  :

  ((((2•3x3) + 11x2) - 9x) + 12)    
  —————————————————————————————— -  1
               -3x                  

Step  3  :

            6x3 + 11x2 - 9x + 12
 Simplify   ————————————————————
                    -3x         

Checking for a perfect cube :

 3.1    6x³ + 11x² - 9x + 12  is not a perfect cube

Trying to factor by pulling out :

 3.2      Factoring:  6x³ + 11x² - 9x + 12 

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

Group 1:  -9x + 12 
Group 2:  6x³ + 11x² 

Pull out from each group separately :

Group 1:   (3x - 4) • (-3)
Group 2:   (6x + 11) • (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 :

 3.3    Find roots (zeroes) of :       F(x) = 6x³ + 11x² - 9x + 12
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  6  and the Trailing Constant is  12.

 The factor(s) are:

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

 Let us test ....

Note - For tidiness, printing of 19 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Polynomial Long Division :

 3.4    Polynomial Long Division
Dividing :  6x³ + 11x² - 9x + 12 
                              ("Dividend")
By         :    -3x    ("Divisor")

Quotient :  -2x² - 3x + 3 
Remainder :  2x² + 12 

Equation at the end of step  3  :

  (6x3 + 11x2 - 9x + 12)    
  —————————————————————— -  1
           -3x              

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  -3x  as the denominator :

         1     1 • -3x
    1 =  —  =  ———————
         1       -3x  

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 (6x3+11x2-9x+12) • -1 - (3x)     -6x3 - 11x2 + 6x - 12
 ————————————————————————————  =  —————————————————————
              3x                           3x          

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   -6x³ - 11x² + 6x - 12  = 

  -1 • (6x³ + 11x² - 6x + 12) 

Checking for a perfect cube :

 5.2    6x³ + 11x² - 6x + 12  is not a perfect cube

Trying to factor by pulling out :

 5.3      Factoring:  6x³ + 11x² - 6x + 12 

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

Group 1:  11x² + 12 
Group 2:  6x³ - 6x 

Pull out from each group separately :

Group 1:   (11x² + 12) • (1)
Group 2:   (x² - 1) • (6x)

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(x) = 6x³ + 11x² - 6x + 12

     See theory in step 3.3
In this case, the Leading Coefficient is  6  and the Trailing Constant is  12.

 The factor(s) are:

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

 Let us test ....

Note - For tidiness, printing of 19 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  -6x3 - 11x2 + 6x - 12
  —————————————————————
           3x