Solution - Nonlinear equations

Step by step solution :

Step  1  :

Trying to factor as a Difference of Cubes:

 1.1      Factoring:  x³-42 

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 :  42  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) = x³-42
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  -42.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,6 ,7 ,14 ,21 ,42

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

  x3 - 42  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    x³-42 = 0 

 Add  42  to both sides of the equation : 
                      x³ = 42
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ 42  

 The equation has one real solution
This solution is  x = ∛42 = 3.4760

One solution was found :