Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            12
 Simplify   ——
            x2

Equation at the end of step  1  :

         1            12
  (((x+————)+x)-12)-((——+5x)-24)
       (x2)           x2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          5x     5x • x2
    5x =  ——  =  ———————
          1        x2   

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 :

 2.2       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:

 12 + 5x • x2     5x3 + 12
 ————————————  =  ————————
      x2             x2   

Equation at the end of step  2  :

         1           (5x3+12)
  (((x+————)+x)-12)-(————————-24)
       (x2)             x2   

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          24     24 • x2
    24 =  ——  =  ———————
          1        x2   

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  5x³ + 12 

Theory : A sum 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 :  5  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = 5x³ + 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  5  and the Trailing Constant is  12.

 The factor(s) are:

of the Leading Coefficient :  1,5
 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

Adding fractions that have a common denominator :

 3.4       Adding up the two equivalent fractions

 (5x3+12) - (24 • x2)     5x3 - 24x2 + 12
 ————————————————————  =  ———————————————
          x2                    x2       

Equation at the end of step  3  :

         1          (5x3-24x2+12)
  (((x+————)+x)-12)-—————————————
       (x2)              x2      
 Step  4  :
             1
 Simplify   ——
            x2

Equation at the end of step  4  :

           1                  (5x3 - 24x2 + 12)
  (((x +  ——) +  x) -  12) -  —————————————————
          x2                         x2        

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  x²  as the denominator :

          x     x • x2
     x =  —  =  ——————
          1       x2  

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 x • x2 + 1     x3 + 1
 ——————————  =  ——————
     x2           x2  

Equation at the end of step  5  :

    (x3 + 1)                 (5x3 - 24x2 + 12)
  ((———————— +  x) -  12) -  —————————————————
       x2                           x2        

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

         x     x • x2
    x =  —  =  ——————
         1       x2  

Trying to factor as a Sum of Cubes :

 6.2      Factoring:  x³ + 1 

Check :  1  is the cube of   1 
Check :  x³ is the cube of   x¹

Factorization is :
             (x + 1)  •  (x² - x + 1) 

Trying to factor by splitting the middle term

 6.3     Factoring  x² - x + 1 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 1 = 1 

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   -1 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

 6.4       Adding up the two equivalent fractions

 (x+1) • (x2-x+1) + x • x2     2x3 + 1
 —————————————————————————  =  ———————
            x2                   x2   

Equation at the end of step  6  :

   (2x3 + 1)           (5x3 - 24x2 + 12)
  (————————— -  12) -  —————————————————
      x2                      x2        

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          12     12 • x2
    12 =  ——  =  ———————
          1        x2   

Trying to factor as a Sum of Cubes :

 7.2      Factoring:  2x³ + 1 

Check :  2  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 7.3    Find roots (zeroes) of :       F(x) = 2x³ + 1

     See theory in step 3.3
In this case, the Leading Coefficient is  2  and the Trailing Constant is  1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 7.4       Adding up the two equivalent fractions

 (2x3+1) - (12 • x2)     2x3 - 12x2 + 1
 ———————————————————  =  ——————————————
         x2                    x2      

Equation at the end of step  7  :

  (2x3 - 12x2 + 1)    (5x3 - 24x2 + 12)
  ———————————————— -  —————————————————
         x2                  x2        

Step  8  :

Polynomial Roots Calculator :

 8.1    Find roots (zeroes) of :       F(x) = 2x³-12x²+1

     See theory in step 3.3
In this case, the Leading Coefficient is  2  and the Trailing Constant is  1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Polynomial Roots Calculator :

 8.2    Find roots (zeroes) of :       F(x) = 5x³-24x²+12

     See theory in step 3.3
In this case, the Leading Coefficient is  5  and the Trailing Constant is  12.

 The factor(s) are:

of the Leading Coefficient :  1,5
 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

Adding fractions which have a common denominator :

 8.3       Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 (2x3-12x2+1) - ((5x3-24x2+12))      -3x3 + 12x2 - 11
 ——————————————————————————————  =  ————————————————
               x2                          x2       

Step  9  :

Pulling out like terms :

 9.1     Pull out like factors :

   -3x³ + 12x² - 11  =   -1 • (3x³ - 12x² + 11) 

Polynomial Roots Calculator :

 9.2    Find roots (zeroes) of :       F(x) = 3x³ - 12x² + 11

     See theory in step 3.3
In this case, the Leading Coefficient is  3  and the Trailing Constant is  11.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,11

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

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