Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  
 Step  2  :

Equation at the end of step  2  :

Step  3  :

            10y3 - 9y2 + 6y - 10
 Simplify   ————————————————————
                   2y + 3       

Checking for a perfect cube :

 3.1    10y³ - 9y² + 6y - 10  is not a perfect cube

Trying to factor by pulling out :

 3.2      Factoring:  10y³ - 9y² + 6y - 10 

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

Group 1:  6y - 10 
Group 2:  10y³ - 9y² 

Pull out from each group separately :

Group 1:   (3y - 5) • (2)
Group 2:   (10y - 9) • (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 :

 3.3    Find roots (zeroes) of :       F(y) = 10y³ - 9y² + 6y - 10
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  10  and the Trailing Constant is  -10.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,5 ,10
 of the Trailing Constant :  1 ,2 ,5 ,10

 Let us test ....

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Polynomial Long Division :

 3.4    Polynomial Long Division
Dividing :  10y³ - 9y² + 6y - 10 
                              ("Dividend")
By         :    2y + 3    ("Divisor")

Quotient :  5y² - 12y + 21 
Remainder :  -73 

Final result :

  10y3 - 9y2 + 6y - 10
  ————————————————————
         2y + 3