Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x8"   was replaced by   "x^8". 

Rearrange:

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

                     3*x^8-(5*x^4)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (3 • (x8)) -  5x4  = 0 
 Step  2  :

Equation at the end of step  2  :

  3x8 -  5x4  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   3x⁸ - 5x⁴  =   x⁴ • (3x⁴ - 5) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  3x⁴ - 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 :  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 - 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  3  and the Trailing Constant is  -5.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,5

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  4  :

  x4 • (3x4 - 5)  = 0 

Step  5  :

Theory - Roots of a product :

 5.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 5.2      Solve  :    x⁴ = 0 

  Solution is  x⁴ = 0

Solving a Single Variable Equation :

 5.3      Solve  :    3x⁴-5 = 0 

 Add  5  to both sides of the equation : 
                      3x⁴ = 5
Divide both sides of the equation by 3:
                     x⁴ = 5/3 = 1.667
                     x  =  ∜ 5/3  

 The equation has two real solutions  
 These solutions are  x = ∜ 1.667 = ± 1.13622  
 

Three solutions were found :

1.  x = ∜ 1.667 = ± 1.13622
2.  x⁴ = 0