Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (54 • (y2)) -  (2•3y6)
 Step  2  :

Equation at the end of step  2  :

  (2•33y2) -  (2•3y6)

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   54y² - 6y⁶  =   -6y² • (y⁴ - 9) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  y⁴ - 9 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 9 is the square of 3
Check :  y⁴  is the square of  y² 

Factorization is :       (y² + 3)  •  (y² - 3) 

Polynomial Roots Calculator :

 4.3    Find roots (zeroes) of :       F(y) = y² + 3
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  1  and the Trailing Constant is  3.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 4.4      Factoring:  y² - 3 

Check : 3 is not a square !!

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

Final result :

  -6y2 • (y2 + 3) • (y2 - 3)