Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x5"   was replaced by   "x^5". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (2x5 • x) -  9  = 0 
 Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  2x⁶-9 

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 :  2  is not a square !!

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

Polynomial Roots Calculator :

 2.2    Find roots (zeroes) of :       F(x) = 2x⁶-9
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  -9.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,3 ,9

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  2  :

  2x6 - 9  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    2x⁶-9 = 0 

 Add  9  to both sides of the equation : 
                      2x⁶ = 9
Divide both sides of the equation by 2:
                     x⁶ = 9/2 = 4.500
                     x  =  6th root of (9/2) 

 The equation has two real solutions  
 These solutions are  x = 6th root of ( 4.500) = ± 1.28490  
 

Two solutions were found :