Solution - Adding, subtracting and finding the least common multiple

Step  1  :

              4  
 Simplify   —————
            x - 5

Equation at the end of step  1  :

     5                    4 
  (—————+(50•((x2)-25)))-———
   (x+5)                 x-5

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  x²-25 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 25 is the square of 5
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 5)  •  (x - 5) 

Equation at the end of step  2  :

     5                    4 
  (—————+50•(x+5)•(x-5))-———
   (x+5)                 x-5

Step  3  :

              5  
 Simplify   —————
            x + 5

Equation at the end of step  3  :

    5                   4 
  (———+50•(x+5)•(x-5))-———
   x+5                 x-5

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  (x+5)  as the denominator :

                              50 • (x + 5) • (x - 5)     50 • (x + 5) • (x - 5) • (x + 5)
    50 • (x + 5) • (x - 5) =  ——————————————————————  =  ————————————————————————————————
                                        1                            (x + 5)             

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 :

 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:

 5 + 50 • (x+5) • (x-5) • (x+5)     50x3 + 250x2 - 1250x - 6245
 ——————————————————————————————  =  ———————————————————————————
           1 • (x+5)                        1 • (x + 5)        

Equation at the end of step  4  :

  (50x3 + 250x2 - 1250x - 6245)      4  
  ————————————————————————————— -  —————
           1 • (x + 5)             x - 5

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   50x³ + 250x² - 1250x - 6245  = 

  5 • (10x³ + 50x² - 250x - 1249) 

Checking for a perfect cube :

 6.2    10x³ + 50x² - 250x - 1249  is not a perfect cube

Trying to factor by pulling out :

 6.3      Factoring:  10x³ + 50x² - 250x - 1249 

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

Group 1:  -250x - 1249 
Group 2:  50x² + 10x³ 

Pull out from each group separately :

Group 1:   (250x + 1249) • (-1)
Group 2:   (x + 5) • (10x²)

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) = 10x³ + 50x² - 250x - 1249
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  10  and the Trailing Constant is  -1249.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,5 ,10
 of the Trailing Constant :  1 ,1249

 Let us test ....

Polynomial Roots Calculator found no rational roots

Calculating the Least Common Multiple :

 6.5    Find the Least Common Multiple

      The left denominator is :       x + 5 

      The right denominator is :       x - 5 

      Least Common Multiple:
      (x + 5) • (x - 5) 

Calculating Multipliers :

 6.6    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = x - 5

   Right_M = L.C.M / R_Deno = x + 5

Making Equivalent Fractions :

 6.7      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      5 • (10x3+50x2-250x-1249) • (x-5)
   ——————————————————  =   —————————————————————————————————
         L.C.M                       (x+5) • (x-5)          

   R. Mult. • R. Num.        4 • (x+5)  
   ——————————————————  =   —————————————
         L.C.M             (x+5) • (x-5)

Adding fractions that have a common denominator :

 6.8       Adding up the two equivalent fractions

 5 • (10x3+50x2-250x-1249) • (x-5) - (4 • (x+5))     50x4 - 2500x2 + x + 31205
 ———————————————————————————————————————————————  =  —————————————————————————
                  (x+5) • (x-5)                          (x + 5) • (x - 5)    

Checking for a perfect cube :

 6.9    50x⁴ - 2500x² + x + 31205  is not a perfect cube

Trying to factor by pulling out :

 6.10      Factoring:  50x⁴ - 2500x² + x + 31205 

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

Group 1:  x + 31205 
Group 2:  50x⁴ - 2500x² 

Pull out from each group separately :

Group 1:   (x + 31205) • (1)
Group 2:   (x² - 50) • (50x²)

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 :

 6.11    Find roots (zeroes) of :       F(x) = 50x⁴ - 2500x² + x + 31205

     See theory in step 6.4
In this case, the Leading Coefficient is  50  and the Trailing Constant is  31205.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,5 ,10 ,25 ,50
 of the Trailing Constant :  1 ,5 ,79 ,395 ,6241 ,31205

 Let us test ....

Note - For tidiness, printing of 43 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  50x4 - 2500x2 + x + 31205
  —————————————————————————
      (x + 5) • (x - 5)