Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (z2) -  24z64
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   z² - 16z⁶⁴  =   -z² • (16z⁶² - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  16z⁶² - 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 :  16  is the square of  4 
Check : 1 is the square of 1
Check :  z⁶²  is the square of  z³¹ 

Factorization is :       (4z³¹ + 1)  •  (4z³¹ - 1) 

Final result :

  -z2 • (4z31 + 1) • (4z31 - 1)