Solution - Finding the roots of polynomials

Step  1  :

              6  
 Simplify   —————
            x - 2

Equation at the end of step  1  :

                         6 
  (((x3)-(6•(x2)))+11x)-———
                        x-2
 Step  2  :

Equation at the end of step  2  :

                                  6  
  (((x3) -  (2•3x2)) +  11x) -  —————
                                x - 2

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  (x-2)  as the denominator :

                       x3 - 6x2 + 11x     (x3 - 6x2 + 11x) • (x - 2)
     x3 - 6x2 + 11x =  ——————————————  =  ——————————————————————————
                             1                     (x - 2)          

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 :

   x³ - 6x² + 11x  =   x • (x² - 6x + 11) 

Trying to factor by splitting the middle term

 4.2     Factoring  x² - 6x + 11 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -6x  its coefficient is  -6 .
The last term, "the constant", is  +11 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 11 = 11 

Step-2 : Find two factors of  11  whose sum equals the coefficient of the middle term, which is   -6 .

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 • (x2-6x+11) • (x-2) - (6)     x4 - 8x3 + 23x2 - 22x - 6 
 ————————————————————————————  =  —————————————————————————
          1 • (x-2)                      1 • (x - 2)       

Polynomial Roots Calculator :

 4.4    Find roots (zeroes) of :       F(x) = x⁴ - 8x³ + 23x² - 22x - 6
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  1  and the Trailing Constant is  -6.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,6

 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
   x⁴ - 8x³ + 23x² - 22x - 6 
can be divided with  x - 3 

Polynomial Long Division :

 4.5    Polynomial Long Division
Dividing :  x⁴ - 8x³ + 23x² - 22x - 6 
                              ("Dividend")
By         :    x - 3    ("Divisor")

Quotient :  x³-5x²+8x+2  Remainder:  0 

Polynomial Roots Calculator :

 4.6    Find roots (zeroes) of :       F(x) = x³-5x²+8x+2

     See theory in step 4.4
In this case, the Leading Coefficient is  1  and the Trailing Constant is  2.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  (x3 - 5x2 + 8x + 2) • (x - 3)
  —————————————————————————————
              x - 2