Solution - Simplification or other simple results

Step  1  :

            4
 Simplify   —
            1

Equation at the end of step  1  :

  (6 • (x4)) -  4
 Step  2  :

Equation at the end of step  2  :

  (2•3x4) -  4

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   6x⁴ - 4  =   2 • (3x⁴ - 2) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  3x⁴ - 2 

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 :  3  is not a square !!

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

Polynomial Roots Calculator :

 4.3    Find roots (zeroes) of :       F(x) = 3x⁴ - 2
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  3  and the Trailing Constant is  -2.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  2 • (3x4 - 2)