Solution - Polynomial long division

Step  1  :

Equation at the end of step  1  :

  (((z3) -  3z2) +  3z) -  1

Step  2  :

Checking for a perfect cube :

 2.1    z³-3z²+3z-1  is not a perfect cube

Trying to factor by pulling out :

 2.2      Factoring:  z³-3z²+3z-1 

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

Group 1:  3z-1 
Group 2:  -3z²+z³ 

Pull out from each group separately :

Group 1:   (3z-1) • (1)
Group 2:   (z-3) • (z²)

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 :

 2.3    Find roots (zeroes) of :       F(z) = z³-3z²+3z-1
Polynomial Roots Calculator is a set of methods aimed at finding values of  z  for which   F(z)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  z  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  -1.

 The factor(s) are:

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

 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
   z³-3z²+3z-1 
can be divided with  z-1 

Polynomial Long Division :

 2.4    Polynomial Long Division
Dividing :  z³-3z²+3z-1 
                              ("Dividend")
By         :    z-1    ("Divisor")

Quotient :  z²-2z+1  Remainder:  0 

Trying to factor by splitting the middle term

 2.5     Factoring  z²-2z+1 

The first term is,  z²  its coefficient is  1 .
The middle term is,  -2z  its coefficient is  -2 .
The last term, "the constant", is  +1 

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

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   -2 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -1  and  -1 
                     z² - 1z - 1z - 1

Step-4 : Add up the first 2 terms, pulling out like factors :
                    z • (z-1)
              Add up the last 2 terms, pulling out common factors :
                     1 • (z-1)
Step-5 : Add up the four terms of step 4 :
                    (z-1)  •  (z-1)
             Which is the desired factorization

Multiplying Exponential Expressions :

 2.6    Multiply  (z-1)  by  (z-1) 

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

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

Multiplying Exponential Expressions :

 2.7    Multiply  (z-1)²   by  (z-1) 

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

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

Final result :

  (z - 1)3