Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2". 

Step  1  :

Equation at the end of step  1  :

  (x2) -  22x24
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² - 4x²⁴  =   -x² • (4x²² - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  4x²² - 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 :  x²²  is the square of  x¹¹ 

Factorization is :       (2x¹¹ + 1)  •  (2x¹¹ - 1) 

Final result :

  -x2 • (2x11 + 1) • (2x11 - 1)