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 :

                     x^3-(105)=0 

Step by step solution :

Step  1  :

Trying to factor as a Difference of Cubes:

 1.1      Factoring:  x³-105 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  105  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 1.2    Find roots (zeroes) of :       F(x) = x³-105
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  -105.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 ,5 ,7 ,15 ,21 ,35 ,105

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

  x3 - 105  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    x³-105 = 0 

 Add  105  to both sides of the equation : 
                      x³ = 105
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ 105  

 The equation has one real solution
This solution is  x = ∛105 = 4.7177

One solution was found :