Solution - Simplification or other simple results

Step  1  :

Trying to factor as a Difference of Cubes:

 1.1      Factoring:  x³-1 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  1  is the cube of   1 
Check :  x³ is the cube of   x¹

Factorization is :
             (x - 1)  •  (x² + x + 1) 

Trying to factor by splitting the middle term

 1.2     Factoring  x² + x + 1 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +x  its coefficient is  1 .
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   1 .

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

Trying to factor as a Difference of Cubes:

 1.3      Factoring:  x³-1 

Check :  1  is the cube of   1 
Check :  x³ is the cube of   x¹

Factorization is :
             (x - 1)  •  (x² + x + 1) 

Trying to factor by splitting the middle term

 1.4     Factoring  x² + x + 1 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +x  its coefficient is  1 .
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   1 .

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

Multiplying Exponential Expressions :

 1.5    Multiply  (x-1)  by  (x-1) 

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

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

Multiplying Exponential Expressions :

 1.6    Multiply  (x²+x+1)  by  (x²+x+1) 

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

In our case, the common base is  (x²+x+1)  and the exponents are :
          1 , as  (x²+x+1)  is the same number as  (x²+x+1)¹ 
 and   1 , as  (x²+x+1)  is the same number as  (x²+x+1)¹ 
The product is therefore,  (x²+x+1)⁽¹⁺¹⁾ = (x²+x+1)² 

Final result :

  (x - 1)2 • (x2 + x + 1)2