Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "y1"   was replaced by   "y^1". 

Step  1  :

Equation at the end of step  1  :

  (4 • (y2)) -  24y16
 Step  2  :

Equation at the end of step  2  :

  22y2 -  24y16

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   4y² - 16y¹⁶  =   -4y² • (4y¹⁴ - 1) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  4y¹⁴ - 1 

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 :  4  is the square of  2 
Check : 1 is the square of 1
Check :  y¹⁴  is the square of  y⁷ 

Factorization is :       (2y⁷ + 1)  •  (2y⁷ - 1) 

Polynomial Roots Calculator :

 4.3    Find roots (zeroes) of :       F(y) = 2y⁷ + 1
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  1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Polynomial Roots Calculator :

 4.4    Find roots (zeroes) of :       F(y) = 2y⁷ - 1

     See theory in step 4.3
In this case, the Leading Coefficient is  2  and the Trailing Constant is  -1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  -4y2 • (2y7 + 1) • (2y7 - 1)