Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

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

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     2*x^6-(10)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  2x6 -  10  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   2x⁶ - 10  =   2 • (x⁶ - 5) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  x⁶ - 5 

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 : 5 is not a square !!

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

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = x⁶ - 5
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  -5.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,5

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  2 • (x6 - 5)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    2   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.2      Solve  :    x⁶-5 = 0 

 Add  5  to both sides of the equation : 
                      x⁶ = 5
                     x  =  6th root of (5) 

 The equation has two real solutions  
 These solutions are  x = ± 6th root of 5 = ± 1.3077  
 

Two solutions were found :