Solution - Reducing fractions to their lowest terms

Step  1  :

            123
 Simplify   ———
            y2 

Equation at the end of step  1  :

   123           
  (——— +  8y) -  20
   y2            

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  y²  as the denominator :

          8y     8y • y2
    8y =  ——  =  ———————
          1        y2   

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:

 123 + 8y • y2     8y3 + 123
 —————————————  =  —————————
      y2              y2    

Equation at the end of step  2  :

  (8y3 + 123)    
  ——————————— -  20
      y2         

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  y²  as the denominator :

          20     20 • y2
    20 =  ——  =  ———————
          1        y2   

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  8y³ + 123 

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 :  8  is the cube of  2 

Check :  123  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(y) = 8y³ + 123
Polynomial Roots Calculator is a set of methods aimed at finding values of  y  for which   F(y)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  y  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  8  and the Trailing Constant is  123.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4 ,8
 of the Trailing Constant :  1 ,3 ,41 ,123

 Let us test ....

Note - For tidiness, printing of 27 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

 (8y3+123) - (20 • y2)     8y3 - 20y2 + 123
 —————————————————————  =  ————————————————
          y2                      y2       

Polynomial Roots Calculator :

 3.5    Find roots (zeroes) of :       F(y) = 8y³ - 20y² + 123

     See theory in step 3.3
In this case, the Leading Coefficient is  8  and the Trailing Constant is  123.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4 ,8
 of the Trailing Constant :  1 ,3 ,41 ,123

 Let us test ....

Note - For tidiness, printing of 27 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  8y3 - 20y2 + 123
  ————————————————
         y2