Solution - Polynomial long division

Step  1  :

                1   
 Simplify   ————————
            (s - 2)5

Equation at the end of step  1  :

                1   
  l -  (1 • ————————)
            (s - 2)5

Step  2  :

Equation at the end of step  2  :

           1   
  l -  ————————
       (s - 2)5

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  (s-2)⁵   as the denominator :

          l     l • (s - 2)5
     l =  —  =  ————————————
          1       (s - 2)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 :

 3.2       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:

 l • (s-2)5 - (1)     ls5 - 10ls4 + 40ls3 - 80ls2 + 80ls - 32l - 1
 ————————————————  =  ————————————————————————————————————————————
    1 • (s-2)5          1 • (s5 - 10s4 + 40s3 - 80s2 + 80s - 32)  

Trying to factor by pulling out :

 3.3      Factoring:  s⁵ - 10s⁴ + 40s³ - 80s² + 80s - 32 

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

Group 1:  -10s⁴ + s⁵ 
Group 2:  -80s² + 40s³ 
Group 3:  80s - 32 

Pull out from each group separately :

Group 1:   (s - 10) • (s⁴)
Group 2:   (s - 2) • (40s²)
Group 3:   (5s - 2) • (16)


Looking for common sub-expressions :

Group 1:   (s - 10) • (s⁴)
Group 3:   (5s - 2) • (16)
Group 2:   (s - 2) • (40s²)

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 :

 3.4    Find roots (zeroes) of :       F(s) = s⁵ - 10s⁴ + 40s³ - 80s² + 80s - 32
Polynomial Roots Calculator is a set of methods aimed at finding values of  s  for which   F(s)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  s  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  -32.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32

 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
   s⁵ - 10s⁴ + 40s³ - 80s² + 80s - 32 
can be divided with  s - 2 

Polynomial Long Division :

 3.5    Polynomial Long Division
Dividing :  s⁵ - 10s⁴ + 40s³ - 80s² + 80s - 32 
                              ("Dividend")
By         :    s - 2    ("Divisor")

Quotient :  s⁴-8s³+24s²-32s+16  Remainder:  0 

Polynomial Roots Calculator :

 3.6    Find roots (zeroes) of :       F(s) = s⁴-8s³+24s²-32s+16

     See theory in step 3.4
In this case, the Leading Coefficient is  1  and the Trailing Constant is  16.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16

 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
   s⁴-8s³+24s²-32s+16 
can be divided with  s-2 

Polynomial Long Division :

 3.7    Polynomial Long Division
Dividing :  s⁴-8s³+24s²-32s+16 
                              ("Dividend")
By         :    s-2    ("Divisor")

Quotient :  s³-6s²+12s-8  Remainder:  0 

Polynomial Roots Calculator :

 3.8    Find roots (zeroes) of :       F(s) = s³-6s²+12s-8

     See theory in step 3.4
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -8.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8

 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
   s³-6s²+12s-8 
can be divided with  s-2 

Polynomial Long Division :

 3.9    Polynomial Long Division
Dividing :  s³-6s²+12s-8 
                              ("Dividend")
By         :    s-2    ("Divisor")

Quotient :  s²-4s+4  Remainder:  0 

Trying to factor by splitting the middle term

 3.10     Factoring  s²-4s+4 

The first term is,  s²  its coefficient is  1 .
The middle term is,  -4s  its coefficient is  -4 .
The last term, "the constant", is  +4 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 4 = 4 

Step-2 : Find two factors of  4  whose sum equals the coefficient of the middle term, which is   -4 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -2  and  -2 
                     s² - 2s - 2s - 4

Step-4 : Add up the first 2 terms, pulling out like factors :
                    s • (s-2)
              Add up the last 2 terms, pulling out common factors :
                    2 • (s-2)
Step-5 : Add up the four terms of step 4 :
                    (s-2)  •  (s-2)
             Which is the desired factorization

Multiplying Exponential Expressions :

 3.11    Multiply  (s-2)  by  (s-2) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (s-2)  and the exponents are :
          1 , as  (s-2)  is the same number as  (s-2)¹ 
 and   1 , as  (s-2)  is the same number as  (s-2)¹ 
The product is therefore,  (s-2)⁽¹⁺¹⁾ = (s-2)² 

Multiplying Exponential Expressions :

 3.12    Multiply  (s-2)²   by  (s-2) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (s-2)  and the exponents are :
          2
 and   1 , as  (s-2)  is the same number as  (s-2)¹ 
The product is therefore,  (s-2)⁽²⁺¹⁾ = (s-2)³ 

Multiplying Exponential Expressions :

 3.13    Multiply  (s-2)³   by  (s-2) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (s-2)  and the exponents are :
          3
 and   1 , as  (s-2)  is the same number as  (s-2)¹ 
The product is therefore,  (s-2)⁽³⁺¹⁾ = (s-2)⁴ 

Multiplying Exponential Expressions :

 3.14    Multiply  (s-2)⁴   by  (s-2) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (s-2)  and the exponents are :
          4
 and   1 , as  (s-2)  is the same number as  (s-2)¹ 
The product is therefore,  (s-2)⁽⁴⁺¹⁾ = (s-2)⁵ 

Final result :

  ls5 - 10ls4 + 40ls3 - 80ls2 + 80ls - 32l - 1 
  ————————————————————————————————————————————
                    (s - 2)5