Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

     (((x2)-2x)-35)  
  ——————————————————— ÷ (22x3-9x) ÷ (7x-49)
  ((2•(x3))-(3•(x2)))
 Step  2  :

Equation at the end of step  2  :

  (((x2)-2x)-35)
  —————————————— ÷ (4x3-9x) ÷ (7x-49)
  ((2•(x3))-3x2)
 Step  3  :

Equation at the end of step  3  :

  (((x2)-2x)-35)
  —————————————— ÷ (4x3-9x) ÷ (7x-49)
    (2x3-3x2)   

Step  4  :

            x2 - 2x - 35
 Simplify   ————————————
             2x3 - 3x2  

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   2x³ - 3x²  =   x² • (2x - 3) 

Trying to factor by splitting the middle term

 5.2     Factoring  x² - 2x - 35 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -2x  its coefficient is  -2 .
The last term, "the constant", is  -35 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -35 = -35 

Step-2 : Find two factors of  -35  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,  -7  and  5 
                     x² - 7x + 5x - 35

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-7)
              Add up the last 2 terms, pulling out common factors :
                    5 • (x-7)
Step-5 : Add up the four terms of step 4 :
                    (x+5)  •  (x-7)
             Which is the desired factorization

Equation at the end of step  5  :

  (x + 5) • (x - 7)
  ————————————————— ÷ (4x3 - 9x) ÷ (7x - 49)
    x2 • (2x - 3)  

Step  6  :

         (x+5)•(x-7)      
 Divide  ———————————  by  4x3-9x
          x2•(2x-3)       

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   4x³ - 9x  =   x • (4x² - 9) 

Trying to factor as a Difference of Squares :

 7.2      Factoring:  4x² - 9 

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 :  4  is the square of  2 
Check : 9 is the square of 3
Check :  x²  is the square of  x¹ 

Factorization is :       (2x + 3)  •  (2x - 3) 

Multiplying exponential expressions :

 7.3    x² multiplied by x¹ = x^{(2 + 1)} = x³

Multiplying Exponential Expressions :

 7.4    Multiply  (2x - 3)  by  (2x - 3) 

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

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

Equation at the end of step  7  :

      (x + 5) • (x - 7)    
  ————————————————————————— ÷ (7x - 49)
  x3 • (2x - 3)2 • (2x + 3)

Step  8  :

            (x+5)•(x-7)         
 Divide  —————————————————  by  7x-49
         x3•(2x-3)2•(2x+3)      

Step  9  :

Pulling out like terms :

 9.1     Pull out like factors :

   7x - 49  =   7 • (x - 7) 

Canceling Out :

 9.2    Cancel out  (x - 7)  which appears on both sides of the fraction line.

Final result :

             x + 5          
  ——————————————————————————
  7x3 • (2x - 3)2 • (2x + 3)