Solution - Equations reducible to quadratic form

Step by step solution :

Step  1  :

Trying to factor as a Difference of Cubes:

 1.1      Factoring:  x²⁷-27 

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 :  27  is the cube of   3 
Check :  x²⁷ is the cube of   x⁹

Factorization is :
             (x⁹ - 3)  •  (x¹⁸ + 3x⁹ + 9) 

Trying to factor as a Difference of Cubes:

 1.2      Factoring:  x⁹ - 3 

Check :  3  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 1.3    Find roots (zeroes) of :       F(x) = x⁹ - 3
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  -3.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor by splitting the middle term

 1.4     Factoring  x¹⁸ + 3x⁹ + 9 

The first term is,  x¹⁸  its coefficient is  1 .
The middle term is,  +3x⁹  its coefficient is  3 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 9 = 9 

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   3 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  1  :

  (x9 - 3) • (x18 + 3x9 + 9)  = 0 

Step  2  :

Theory - Roots of a product :

 2.1    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 :

 2.2      Solve  :    x⁹-3 = 0 

 Add  3  to both sides of the equation : 
                      x⁹ = 3
                     x  =  9th root of (3) 

 The equation has one real solution
This solution is  x = 9th root of 3 = 1.1298

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 2.3     Solve   x¹⁸+3x⁹+9 = 0

This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using  w , such that  w = x⁹  transforms the equation into :
 w²+3w+9 = 0

Solving this new equation using the quadratic formula we get two imaginary solutions :
   w = -1.5000 ± 2.5981 i 
Now that we know the value(s) of  w , we can calculate  x  since  x  is the 9 root of   w  

Since we are speaking 9th root, each of the two imaginary solutions of has 9 roots

Tiger finds these roots using de Moivre's Formula

The 9th roots of  -1.500 + 2.598 i   are:

9th roots of  -1.500- 2.598 i  :
 

19 solutions were found :