Solution - Finding the roots of polynomials

Step  1  :

            5
 Simplify   —
            x

Equation at the end of step  1  :

                              5
  ((((5•(x3))+(21•(x2)))-21x)-—)+5
                              x
 Step  2  :

Equation at the end of step  2  :

                            5
  ((((5•(x3))+(3•7x2))-21x)-—)+5
                            x
 Step  3  :

Equation at the end of step  3  :

                                5     
  (((5x3 +  (3•7x2)) -  21x) -  —) +  5
                                x     

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a fraction from a whole

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

                         5x3 + 21x2 - 21x     (5x3 + 21x2 - 21x) • x
     5x3 + 21x2 - 21x =  ————————————————  =  ——————————————————————
                                1                       x           

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  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   5x³ + 21x² - 21x  =   x • (5x² + 21x - 21) 

Trying to factor by splitting the middle term

 5.2     Factoring  5x² + 21x - 21 

The first term is,  5x²  its coefficient is  5 .
The middle term is,  +21x  its coefficient is  21 .
The last term, "the constant", is  -21 

Step-1 : Multiply the coefficient of the first term by the constant   5 • -21 = -105 

Step-2 : Find two factors of  -105  whose sum equals the coefficient of the middle term, which is   21 .

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

Adding fractions that have a common denominator :

 5.3       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 • (5x2+21x-21) • x - (5)     5x4 + 21x3 - 21x2 - 5 
 ——————————————————————————  =  —————————————————————
             x                            x          

Equation at the end of step  5  :

  (5x4 + 21x3 - 21x2 - 5)     
  ——————————————————————— +  5
             x               

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

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

         5     5 • x
    5 =  —  =  —————
         1       x  

Checking for a perfect cube :

 6.2    5x⁴ + 21x³ - 21x² - 5  is not a perfect cube

Trying to factor by pulling out :

 6.3      Factoring:  5x⁴ + 21x³ - 21x² - 5 

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

Group 1:  5x⁴ - 5 
Group 2:  21x³ - 21x² 

Pull out from each group separately :

Group 1:   (x⁴ - 1) • (5)
Group 2:   (x - 1) • (21x²)

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) = 5x⁴ + 21x³ - 21x² - 5
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  5  and the Trailing Constant is  -5.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,5

 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
   5x⁴ + 21x³ - 21x² - 5 
can be divided with  x - 1 

Polynomial Long Division :

 6.5    Polynomial Long Division
Dividing :  5x⁴ + 21x³ - 21x² - 5 
                              ("Dividend")
By         :    x - 1    ("Divisor")

Quotient :  5x³+26x²+5x+5  Remainder:  0 

Polynomial Roots Calculator :

 6.6    Find roots (zeroes) of :       F(x) = 5x³+26x²+5x+5

     See theory in step 6.4
In this case, the Leading Coefficient is  5  and the Trailing Constant is  5.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,5

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 6.7       Adding up the two equivalent fractions

 (5x3+26x2+5x+5) • (x-1) + 5 • x     5x4 + 21x3 - 21x2 + 5x - 5 
 ———————————————————————————————  =  ——————————————————————————
                x                                x             

Polynomial Roots Calculator :

 6.8    Find roots (zeroes) of :       F(x) = 5x⁴ + 21x³ - 21x² + 5x - 5

     See theory in step 6.4
In this case, the Leading Coefficient is  5  and the Trailing Constant is  -5.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,5

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  5x4 + 21x3  21x2 + 5x - 5 
  ——————————————————————————
              x