Solution - Factoring binomials using the difference of squares

Step  1  :

            (r - s)4
 Simplify   ————————
            r3 + s3 

Trying to factor as a Sum of Cubes :

 1.1      Factoring:  r³ + s³ 

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 :  r³ is the cube of   r¹

Check :  s³ is the cube of   s¹

Factorization is :
             (r + s)  •  (r² - rs + s²) 

Trying to factor a multi variable polynomial :

 1.2    Factoring    r² - rs + s² 

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

 Factorization fails

Equation at the end of step  1  :

  ((r2)-(s2))   (((r2)-2rs)+(s2))      (r-s)4     
  ——————————— ÷ —————————————————•————————————————
   ((r-s)2)     (((r2)-rs)+(s2))  (r+s)•(r2-rs+s2)

Step  2  :

            r2 - 2rs + s2
 Simplify   —————————————
            r2 - rs + s2 

Trying to factor a multi variable polynomial :

 2.1    Factoring    r² - 2rs + s² 

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

 Found a factorization  :  (r - s)•(r - s)

Detecting a perfect square :

 2.2    r²  -2rs  +s²  is a perfect square 

 It factors into  (r-s)•(r-s)
which is another way of writing  (r-s)²

How to recognize a perfect square trinomial:  

 • It has three terms  

 • Two of its terms are perfect squares themselves  

 • The remaining term is twice the product of the square roots of the other two terms

Trying to factor a multi variable polynomial :

 2.3    Factoring    r² - rs + s² 

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

 Factorization fails

Equation at the end of step  2  :

  ((r2)-(s2))    (r-s)2       (r-s)4     
  ——————————— ÷ ————————•————————————————
   ((r-s)2)     r2-rs+s2 (r+s)•(r2-rs+s2)

Step  3  :

             r2 - s2
 Simplify   ————————
            (r - s)2

Trying to factor as a Difference of Squares :

 3.1      Factoring:  r² - s² 

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 :  r²  is the square of  r¹ 

Check :  s²  is the square of  s¹ 

Factorization is :       (r + s)  •  (r - s) 

Dividing Exponential Expressions :

 3.2    Divide  (r - s)  by  (r - s)²  

The rule says : To divide exponential expressions which have the same base, subtract their exponents.

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

Note that the quotient has a negative exponent.
Rewrite the quotient under the fraction line changing its exponent to be positive: 1/(r-s)¹

Omit the '1' in the exponent altogether. Anything to the first power is the number itself so there is usually no reason to write down the '1

Equation at the end of step  3  :

  (r+s)    (r-s)2       (r-s)4     
  ————— ÷ ————————•————————————————
   r-s    r2-rs+s2 (r+s)•(r2-rs+s2)

Step  4  :

          r+s         (r-s)2  
 Divide  —————  by  ——————————
         (r-s)      (r2-rs+s2)

 4.1    Dividing fractions

To divide fractions, write the divison as multiplication by the reciprocal of the divisor :

 r + s         (r - s)2           r + s      r2 - rs + s2
———————  ÷  ——————————————   =   ———————  •  ————————————
(r - s)     (r2 - rs + s2)       (r - s)       (r - s)2  

Multiplying Exponential Expressions :

 4.2    Multiply  (r - s)  by  (r - 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  (r-s)  and the exponents are :
          1 , as  (r-s)  is the same number as  (r-s)¹ 
 and   2
The product is therefore,  (r-s)⁽¹⁺²⁾ = (r-s)³ 

Equation at the end of step  4  :

  (r+s)•(r2-rs+s2)      (r-s)4     
  ————————————————•————————————————
       (r-s)3      (r+s)•(r2-rs+s2)

Step  5  :

Canceling Out :

 5.1    Cancel out  (r+s)  which appears on both sides of the fraction line.

Canceling Out :

 5.2    Cancel out  (r²-rs+s²)  which appears on both sides of the fraction line.

Dividing Exponential Expressions :

 5.3    Divide  (r-s)⁴   by  (r-s)³  

The rule says : To divide exponential expressions which have the same base, subtract their exponents.

In our case, the common base is  (r-s)  and the exponents are :
          4
 and   3
The quotient is therefore,  (r-s)^(4-3) = (r-s)¹ 

Omit the '1' in the exponent altogether. Anything to the first power is the number itself so there is usually no reason to write down the '1

Final result :

  r - s