Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-2" was replaced by "^(-2)".

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   e² - e^(-2)  =   e^(-2) • (e⁴ - 1) 

Trying to factor as a Difference of Squares :

 2.2      Factoring:  e⁴ - 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 : 1 is the square of 1
Check :  e⁴  is the square of  e² 

Factorization is :       (e² + 1)  •  (e² - 1) 

Polynomial Roots Calculator :

 2.3    Find roots (zeroes) of :       F(e) = e² + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  e  for which   F(e)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  e  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  1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 2.4      Factoring:  e² - 1 

Check : 1 is the square of 1
Check :  e²  is the square of  e¹ 

Factorization is :       (e + 1)  •  (e - 1) 

Final result :

  e(-2) • (e2 + 1) • (e + 1) • (e - 1)