Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "y5"   was replaced by   "y^5". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  32y2 -  y5  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   9y² - y⁵  =   -y² • (y³ - 9) 

Trying to factor as a Difference of Cubes:

 3.2      Factoring:  y³ - 9 

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

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(y) = y³ - 9
Polynomial Roots Calculator is a set of methods aimed at finding values of  y  for which   F(y)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  y  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  -9.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 ,9

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  -y2 • (y3 - 9)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    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 :

 4.2      Solve  :    -y² = 0 

 Multiply both sides of the equation by (-1) :  y² = 0


 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      y  =  ± √ 0  

 Any root of zero is zero. This equation has one solution which is  y = 0

Solving a Single Variable Equation :

 4.3      Solve  :    y³-9 = 0 

 Add  9  to both sides of the equation : 
                      y³ = 9
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      y  =  ∛ 9  

 The equation has one real solution
This solution is  y = ∛9 = 2.0801

Two solutions were found :

1.  y = ∛9 = 2.0801
2.  y = 0