Solution - Adding, subtracting and finding the least common multiple

Step  1  :

                  5      
 Simplify   —————————————
            x2 - 10x - 25

Trying to factor by splitting the middle term

 1.1     Factoring  x² - 10x - 25 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -10x  its coefficient is  -10 .
The last term, "the constant", is  -25 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -25 = -25 

Step-2 : Find two factors of  -25  whose sum equals the coefficient of the middle term, which is   -10 .

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

Equation at the end of step  1  :

             5      
  x -  —————————————
       x2 - 10x - 25

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  (x²-10x-25)  as the denominator :

          x     x • (x2 - 10x - 25)
     x =  —  =  ———————————————————
          1       (x2 - 10x - 25)  

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

Adding fractions that have a common denominator :

 2.2       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 • (x2-10x-25) - (5)     x3 - 10x2 - 25x - 5
 —————————————————————  =  ———————————————————
    1 • (x2-10x-25)        1 • (x2 - 10x - 25)

Checking for a perfect cube :

 2.3    x³ - 10x² - 25x - 5  is not a perfect cube

Trying to factor by pulling out :

 2.4      Factoring:  x³ - 10x² - 25x - 5 

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

Group 1:  -25x - 5 
Group 2:  -10x² + x³ 

Pull out from each group separately :

Group 1:   (5x + 1) • (-5)
Group 2:   (x - 10) • (x²)

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 :

 2.5    Find roots (zeroes) of :       F(x) = x³ - 10x² - 25x - 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  1  and the Trailing Constant is  -5.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  x3 + 10x2 + 25x + 5
  ———————————————————
     x2 + 10x + 25