Solution - Finding the roots of polynomials

Step  1  :

             9
 Simplify   ——
            v2

Equation at the end of step  1  :

                   9
  ((((3•(v2))-6v)-——)-9v)+18
                  v2
 Step  2  :

Equation at the end of step  2  :

                    9            
  (((3v2 -  6v) -  ——) -  9v) +  18
                   v2            

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  v²  as the denominator :

                 3v2 - 6v     (3v2 - 6v) • v2
     3v2 - 6v =  ————————  =  ———————————————
                    1               v2       

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   3v² - 6v  =   3v • (v - 2) 

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 3v • (v-2) • v2 - (9)     3v4 - 6v3 - 9
 —————————————————————  =  —————————————
          v2                    v2      

Equation at the end of step  4  :

   (3v4 - 6v3 - 9)           
  (——————————————— -  9v) +  18
         v2                  

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  v²  as the denominator :

          9v     9v • v2
    9v =  ——  =  ———————
          1        v2   

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   3v⁴ - 6v³ - 9  =   3 • (v⁴ - 2v³ - 3) 

Polynomial Roots Calculator :

 6.2    Find roots (zeroes) of :       F(v) = v⁴ - 2v³ - 3
Polynomial Roots Calculator is a set of methods aimed at finding values of  v  for which   F(v)=0  

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

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   v⁴ - 2v³ - 3 
can be divided with  v + 1 

Polynomial Long Division :

 6.3    Polynomial Long Division
Dividing :  v⁴ - 2v³ - 3 
                              ("Dividend")
By         :    v + 1    ("Divisor")

Quotient :  v³-3v²+3v-3  Remainder:  0 

Polynomial Roots Calculator :

 6.4    Find roots (zeroes) of :       F(v) = v³-3v²+3v-3

     See theory in step 6.2
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

Adding fractions that have a common denominator :

 6.5       Adding up the two equivalent fractions

 3 • (v3-3v2+3v-3) • (v+1) - (9v • v2)     3v4 - 15v3 - 9
 —————————————————————————————————————  =  ——————————————
                  v2                             v2      

Equation at the end of step  6  :

  (3v4 - 15v3 - 9)    
  ———————————————— +  18
         v2           

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  v²  as the denominator :

          18     18 • v2
    18 =  ——  =  ———————
          1        v2   

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   3v⁴ - 15v³ - 9  =   3 • (v⁴ - 5v³ - 3) 

Polynomial Roots Calculator :

 8.2    Find roots (zeroes) of :       F(v) = v⁴ - 5v³ - 3

     See theory in step 6.2
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

Adding fractions that have a common denominator :

 8.3       Adding up the two equivalent fractions

 3 • (v4-5v3-3) + 18 • v2     3v4 - 15v3 + 18v2 - 9
 ————————————————————————  =  —————————————————————
            v2                         v2          

Step  9  :

Pulling out like terms :

 9.1     Pull out like factors :

   3v⁴ - 15v³ + 18v² - 9  = 

  3 • (v⁴ - 5v³ + 6v² - 3) 

Checking for a perfect cube :

 9.2    v⁴ - 5v³ + 6v² - 3  is not a perfect cube

Trying to factor by pulling out :

 9.3      Factoring:  v⁴ - 5v³ + 6v² - 3 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  6v² - 3 
Group 2:  v⁴ - 5v³ 

Pull out from each group separately :

Group 1:   (2v² - 1) • (3)
Group 2:   (v - 5) • (v³)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 9.4    Find roots (zeroes) of :       F(v) = v⁴ - 5v³ + 6v² - 3

     See theory in step 6.2
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

Final result :

  3 • (v4  5v3 + 6v2 - 3)
  ————————————————————————
             v2