Solution - Finding the roots of polynomials

Step  1  :

             4
 Simplify   ——
            x2

Equation at the end of step  1  :

                            4
  (((((((3•(x3))-(x2))+2x)-——)-3x)+2)•d)•x
                           x2
 Step  2  :

Equation at the end of step  2  :

                     4
  ((((((3x3-x2)+2x)-——)-3x)+2)•d)•x
                    x2

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

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

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

   3x³ - x² + 2x  =   x • (3x² - x + 2) 

Trying to factor by splitting the middle term

 4.2     Factoring  3x² - x + 2 

The first term is,  3x²  its coefficient is  3 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is  +2 

Step-1 : Multiply the coefficient of the first term by the constant   3 • 2 = 6 

Step-2 : Find two factors of  6  whose sum equals the coefficient of the middle term, which is   -1 .

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

Adding fractions that have a common denominator :

 4.3       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 • (3x2-x+2) • x2 - (4)     3x5 - x4 + 2x3 - 4
 ————————————————————————  =  ——————————————————
            x2                        x2        

Equation at the end of step  4  :

     (3x5 - x4 + 2x3 - 4)           
  (((———————————————————— -  3x) +  2) • d) • x
              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   

Checking for a perfect cube :

 5.2    3x⁵ - x⁴ + 2x³ - 4  is not a perfect cube

Trying to factor by pulling out :

 5.3      Factoring:  3x⁵ - x⁴ + 2x³ - 4 

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

Group 1:  2x³ - 4 
Group 2:  3x⁵ - x⁴ 

Pull out from each group separately :

Group 1:   (x³ - 2) • (2)
Group 2:   (3x - 1) • (x⁴)

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 :

 5.4    Find roots (zeroes) of :       F(x) = 3x⁵ - x⁴ + 2x³ - 4
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  -4.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,2 ,4

 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
   3x⁵ - x⁴ + 2x³ - 4 
can be divided with  x - 1 

Polynomial Long Division :

 5.5    Polynomial Long Division
Dividing :  3x⁵ - x⁴ + 2x³ - 4 
                              ("Dividend")
By         :    x - 1    ("Divisor")

Quotient :  3x⁴+2x³+4x²+4x+4  Remainder:  0 

Polynomial Roots Calculator :

 5.6    Find roots (zeroes) of :       F(x) = 3x⁴+2x³+4x²+4x+4

     See theory in step 5.4
In this case, the Leading Coefficient is  3  and the Trailing Constant is  4.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,2 ,4

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.7       Adding up the two equivalent fractions

 (3x4+2x3+4x2+4x+4) • (x-1) - (3x • x2)      3x5 - x4 - x3 - 4
 ——————————————————————————————————————  =  —————————————————
                   x2                              x2        

Equation at the end of step  5  :

    (3x5 - x4 - x3 - 4)    
  ((——————————————————— +  2) • d) • x
            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  

Checking for a perfect cube :

 6.2    3x⁵ - x⁴ - x³ - 4  is not a perfect cube

Trying to factor by pulling out :

 6.3      Factoring:  3x⁵ - x⁴ - x³ - 4 

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

Group 1:  -x³ - 4 
Group 2:  3x⁵ - x⁴ 

Pull out from each group separately :

Group 1:   (x³ + 4) • (-1)
Group 2:   (3x - 1) • (x⁴)

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 :

 6.4    Find roots (zeroes) of :       F(x) = 3x⁵ - x⁴ - x³ - 4

     See theory in step 5.4
In this case, the Leading Coefficient is  3  and the Trailing Constant is  -4.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,2 ,4

 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

 (3x5-x4-x3-4) + 2 • x2      3x5 - x4 - x3 + 2x2 - 4 
 ——————————————————————  =  ———————————————————————
           x2                         x2           

Equation at the end of step  6  :

   (3x5 - x4 - x3 + 2x2 - 4) 
  (————————————————————————— • d) • x
              x2            

Step  7  :

Polynomial Roots Calculator :

 7.1    Find roots (zeroes) of :       F(x) = 3x⁵-x⁴-x³+2x²-4

     See theory in step 5.4
In this case, the Leading Coefficient is  3  and the Trailing Constant is  -4.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,2 ,4

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  7  :

  d • (3x5 - x4 - x3 + 2x2 - 4) 
  ————————————————————————————— • x
               x2              

Step  8  :

Dividing exponential expressions :

 8.1    x¹ divided by x² = x^{(1 - 2)} = x^(-1) = ¹/_x¹ = ¹/_x

Final result :

  d • (3x5 - x4 - x3 + 2x2 - 4) 
  —————————————————————————————
                x