Solution - Reducing fractions to their lowest terms

Step  1  :

            40
 Simplify   ——
            y 

Equation at the end of step  1  :

                         40     
  (((2 • (y2)) +  3y) -  ——) +  5
                         y      
 Step  2  :

Equation at the end of step  2  :

                  40     
  ((2y2 +  3y) -  ——) +  5
                  y      

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

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

                 2y2 + 3y     (2y2 + 3y) • y
     2y2 + 3y =  ————————  =  ——————————————
                    1               y       

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

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   2y² + 3y  =   y • (2y + 3) 

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:

 y • (2y+3) • y - (40)     2y3 + 3y2 - 40
 —————————————————————  =  ——————————————
           y                     y       

Equation at the end of step  4  :

  (2y3 + 3y2 - 40)    
  ———————————————— +  5
         y            

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a whole to a fraction

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

         5     5 • y
    5 =  —  =  —————
         1       y  

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(y) = 2y³ + 3y² - 40
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  2  and the Trailing Constant is  -40.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,2 ,4 ,5 ,8 ,10 ,20 ,40

 Let us test ....

Note - For tidiness, printing of 15 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (2y3+3y2-40) + 5 • y     2y3 + 3y2 + 5y - 40
 ————————————————————  =  ———————————————————
          y                        y         

Checking for a perfect cube :

 5.4    2y³ + 3y² + 5y - 40  is not a perfect cube

Trying to factor by pulling out :

 5.5      Factoring:  2y³ + 3y² + 5y - 40 

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

Group 1:  5y - 40 
Group 2:  2y³ + 3y² 

Pull out from each group separately :

Group 1:   (y - 8) • (5)
Group 2:   (2y + 3) • (y²)

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 :

 5.6    Find roots (zeroes) of :       F(y) = 2y³ + 3y² + 5y - 40

     See theory in step 5.2
In this case, the Leading Coefficient is  2  and the Trailing Constant is  -40.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,2 ,4 ,5 ,8 ,10 ,20 ,40

 Let us test ....

Note - For tidiness, printing of 15 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  2y3 + 3y2 + 5y - 40
  ———————————————————
           y