Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (72 • (x2)) -  2x12
 Step  2  :

Equation at the end of step  2  :

  (23•32x2) -  2x12

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   72x² - 2x¹²  =   -2x² • (x¹⁰ - 36) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  x¹⁰ - 36 

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 : 36 is the square of 6
Check :  x¹⁰  is the square of  x⁵ 

Factorization is :       (x⁵ + 6)  •  (x⁵ - 6) 

Polynomial Roots Calculator :

 4.3    Find roots (zeroes) of :       F(x) = x⁵ + 6
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  1  and the Trailing Constant is  6.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,6

 Let us test ....

Polynomial Roots Calculator found no rational roots

Polynomial Roots Calculator :

 4.4    Find roots (zeroes) of :       F(x) = x⁵ - 6

     See theory in step 4.3
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -6.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,6

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  -2x2 • (x5 + 6) • (x5 - 6)