Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x6"   was replaced by   "x^6". 

Step  1  :

Equation at the end of step  1  :

  (x2) -  22x6
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² - 4x⁶  =   -x² • (4x⁴ - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  4x⁴ - 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 :  x⁴  is the square of  x² 

Factorization is :       (2x² + 1)  •  (2x² - 1) 

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = 2x² + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

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

Trying to factor as a Difference of Squares :

 3.4      Factoring:  2x² - 1 

Check :  2  is not a square !!

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

Final result :

  -x2 • (2x2 + 1) • (2x2 - 1)