Solution - Linear equations with one unknown

Rearrange:

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

                     6*u-(6*(-4*u^5))=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  6u -  (6 • (0 -  22u5))  = 0 

Step  2  :

Multiplying exponents :

 2.1    2¹  multiplied by  2²   = 2^{(1 + 2)} = 2³

Equation at the end of step  2  :

  6u -  ( -23•3u5)  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   24u⁵ + 6u  =   6u • (4u⁴ + 1) 

Polynomial Roots Calculator :

 4.2    Find roots (zeroes) of :       F(u) = 4u⁴ + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  u  for which   F(u)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  u  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  4  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  4  :

  6u • (4u4 + 1)  = 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  :    6u = 0 

 Divide both sides of the equation by 6:
                     u = 0

Solving a Single Variable Equation :

 5.3      Solve  :    4u⁴+1 = 0 

 Subtract  1  from both sides of the equation : 
                      4u⁴ = -1
Divide both sides of the equation by 4:
                     u⁴ = -1/4 = -0.250
                     u  =  ∜ -1/4  

 The equation has no real solutions. It has 4 imaginary, or complex solutions.

                      u=  0.5000 + 0.5000 i 
                      u=  -0.5000 + 0.5000 i 
                      u=  -0.5000 - 0.5000 i 
                      u=  0.5000 - 0.5000 i 

5 solutions were found :

1.   u=  0.5000 - 0.5000 i 
   
2.   u=  -0.5000 - 0.5000 i 
   
3.   u=  -0.5000 + 0.5000 i 
   
4.   u=  0.5000 + 0.5000 i 
   
5.  u = 0