Solution - Finding the roots of polynomials

Step  1  :

             2
 Simplify   ——
            x2

Equation at the end of step  1  :

                   2
  ((((3•(x2))-5x)+——)-3x)+2
                  x2
 Step  2  :

Equation at the end of step  2  :

                    2            
  (((3x2 -  5x) +  ——) -  3x) +  2
                   x2            

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  x²  as the denominator :

                 3x2 - 5x     (3x2 - 5x) • x2
     3x2 - 5x =  ————————  =  ———————————————
                    1               x2       

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 :

   3x² - 5x  =   x • (3x - 5) 

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:

 x • (3x-5) • x2 + 2     3x4 - 5x3 + 2
 ———————————————————  =  —————————————
         x2                   x2      

Equation at the end of step  4  :

   (3x4 - 5x3 + 2)           
  (——————————————— -  3x) +  2
         x2                  

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          3x     3x • x2
    3x =  ——  =  ———————
          1        x2   

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(x) = 3x⁴ - 5x³ + 2
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  3  and the Trailing Constant is  2.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,2

 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
   3x⁴ - 5x³ + 2 
can be divided with  x - 1 

Polynomial Long Division :

 5.3    Polynomial Long Division
Dividing :  3x⁴ - 5x³ + 2 
                              ("Dividend")
By         :    x - 1    ("Divisor")

Quotient :  3x³-2x²-2x-2  Remainder:  0 

Polynomial Roots Calculator :

 5.4    Find roots (zeroes) of :       F(x) = 3x³-2x²-2x-2

     See theory in step 5.2
In this case, the Leading Coefficient is  3  and the Trailing Constant is  -2.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,2

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.5       Adding up the two equivalent fractions

 (3x3-2x2-2x-2) • (x-1) - (3x • x2)     3x4 - 8x3 + 2
 ——————————————————————————————————  =  —————————————
                 x2                          x2      

Equation at the end of step  5  :

  (3x4 - 8x3 + 2)    
  ——————————————— +  2
        x2           

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

         2     2 • x2
    2 =  —  =  ——————
         1       x2  

Polynomial Roots Calculator :

 6.2    Find roots (zeroes) of :       F(x) = 3x⁴ - 8x³ + 2

     See theory in step 5.2
In this case, the Leading Coefficient is  3  and the Trailing Constant is  2.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,2

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 6.3       Adding up the two equivalent fractions

 (3x4-8x3+2) + 2 • x2     3x4 - 8x3 + 2x2 + 2
 ————————————————————  =  ———————————————————
          x2                      x2         

Checking for a perfect cube :

 6.4    3x⁴ - 8x³ + 2x² + 2  is not a perfect cube

Trying to factor by pulling out :

 6.5      Factoring:  3x⁴ - 8x³ + 2x² + 2 

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

Group 1:  2x² + 2 
Group 2:  -8x³ + 3x⁴ 

Pull out from each group separately :

Group 1:   (x² + 1) • (2)
Group 2:   (3x - 8) • (x³)

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 :

 6.6    Find roots (zeroes) of :       F(x) = 3x⁴ - 8x³ + 2x² + 2

     See theory in step 5.2
In this case, the Leading Coefficient is  3  and the Trailing Constant is  2.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,2

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  3x4 - 8x3 + 2x2 + 2
  ———————————————————
          x2