y
Step-by-step explanation:
2•3y3) -  22y2) -  3y) -  —) -  2
                y  Â
STEP
4
:
Rewriting the whole as an Equivalent Fraction
4.1 Â Subtracting a fraction from a whole
Rewrite the whole as a fraction using  y  as the denominator :
           6y3 - 4y2 - 3y   (6y3 - 4y2 - 3y) • y
  6y3 - 4y2 - 3y =  ——————————————  =  ————————————————————
              1           y     Â
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
5
:
Pulling out like terms
5.1 Â Â Pull out like factors :
 6y3 - 4y2 - 3y  =  y • (6y2 - 4y - 3)
Trying to factor by splitting the middle term
5.2   Factoring  6y2 - 4y - 3
The first term is,  6y2  its coefficient is  6 .
The middle term is,  -4y  its coefficient is  -4 .
The last term, "the constant", is  -3
Step-1 : Multiply the coefficient of the first term by the constant  6 • -3 = -18
Step-2 : Find two factors of  -18  whose sum equals the coefficient of the middle term, which is  -4 .
   -18   +   1   =   -17
   -9   +   2   =   -7
   -6   +   3   =   -3
   -3   +   6   =   3
   -2   +   9   =   7
   -1   +   18   =   17
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
5.3 Â Â Â 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 • (6y2-4y-3) • y - (6)   6y4 - 4y3 - 3y2 - 6
————————————————————————  =  ———————————————————
      y              y    Â
Equation at the end of step
5
:
 (6y4 - 4y3 - 3y2 - 6)  Â
 ————————————————————— -  2
      y       Â
STEP
6
:
Rewriting the whole as an Equivalent Fraction :
6.1 Â Subtracting a whole from a fraction
Rewrite the whole as a fraction using  y  as the denominator :
    2   2 • y
  2 =  —  =  —————
    1    y Â
Checking for a perfect cube :
6.2 Â Â 6y4 - 4y3 - 3y2 - 6 Â is not a perfect cube
Trying to factor by pulling out :
6.3 Â Â Â Factoring: Â 6y4 - 4y3 - 3y2 - 6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: Â -3y2 - 6
Group 2: Â 6y4 - 4y3
Pull out from each group separately :
Group 1:  (y2 + 2) • (-3)
Group 2:  (3y - 2) • (2y3)
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 :
6.4 Â Â Find roots (zeroes) of : Â Â Â F(y) = 6y4 - 4y3 - 3y2 - 6
Polynomial Roots Calculator is a set of methods aimed at finding values of  y  for which  F(y)=0 Â
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  6  and the Trailing Constant is  -6.
The factor(s) are:
of the Leading Coefficient : Â 1,2 ,3 ,6
of the Trailing Constant : Â 1 ,2 ,3 ,6
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     1.00  Â
   -1    2     -0.50     -5.88  Â
   -1    3     -0.33     -6.11  Â
   -1    6     -0.17     -6.06  Â
   -2    1     -2.00     110.00  Â
Note - For tidiness, printing of 13 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
6.5 Â Â Â Adding up the two equivalent fractions
(6y4-4y3-3y2-6) - (2 • y)    6y4 - 4y3 - 3y2 - 2y - 6
—————————————————————————  =  ————————————————————————
      y               y      Â
Polynomial Roots Calculator :
6.6 Â Â Find roots (zeroes) of : Â Â Â F(y) = 6y4 - 4y3 - 3y2 - 2y - 6
  See theory in step 6.4
In this case, the Leading Coefficient is  6  and the Trailing Constant is  -6.
The factor(s) are:
of the Leading Coefficient : Â 1,2 ,3 ,6
of the Trailing Constant : Â 1 ,2 ,3 ,6
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     3.00  Â
   -1    2     -0.50     -4.88  Â
   -1    3     -0.33     -5.44  Â
   -1    6     -0.17     -5.73  Â
   -2    1     -2.00     114.00  Â
Note - For tidiness, printing of 13 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Final result :
 6y4 - 4y3 - 3y2 - 2y - 6
 ————————————————————————
      y      Â