Solution - Factoring binomials using the difference of squares

Step  1  :

              y2 
 Simplify   —————
            x - y

Equation at the end of step  1  :

  + y2 
  —————) ÷ (x3 + y3) ÷ (x2 - y2)
  x - y

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  (x-y)  as the denominator :

          x     x • (x - y)
     x =  —  =  ———————————
          1       (x - y)  

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 • (x-y) + y2     x2 - xy + y2
 ——————————————  =  ————————————
   1 • (x-y)        1 • (x - y) 

Equation at the end of step  2  :

  (x2 - xy + y2)
  —————————————— ÷ (x3 + y3) ÷ (x2 - y2)
   1 • (x - y)  

Step  3  :

         x2-xy+y2      
 Divide  ————————  by  x3+y3
         1•(x-y)       

Trying to factor a multi variable polynomial :

 3.1    Factoring    x² - xy + y² 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  x³ + y³ 

Theory : A sum 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 :  x³ is the cube of   x¹

Check :  y³ is the cube of   y¹

Factorization is :
             (x + y)  •  (x² - xy + y²) 

Trying to factor a multi variable polynomial :

 3.3    Factoring    x² - xy + y² 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Canceling Out :

 3.4    Cancel out  (x² - xy + y²)  which appears on both sides of the fraction line.

Equation at the end of step  3  :

          1        
  ————————————————— ÷ (x2 - y2)
  (x - y) • (x + y)

Step  4  :

              1           
 Divide  ———————————  by  x2-y2
         (x-y)•(x+y)      

Trying to factor as a Difference of Squares :

 4.1      Factoring:  x² - y² 

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 :  x²  is the square of  x¹ 

Check :  y²  is the square of  y¹ 

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

Multiplying Exponential Expressions :

 4.2    Multiply  (x - y)  by  (x - y) 

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

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

Multiplying Exponential Expressions :

 4.3    Multiply  (x+y)  by  (x+y) 

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

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

Final result :

           1         
  ———————————————————
  (x + y)2 • (x + y)2