Solution - Power equations

Step  1  :

Equation at the end of step  1  :

  
 Step  2  :

Equation at the end of step  2  :

  
 Step  3  :

Equation at the end of step  3  :

  
 Step  4  :

Equation at the end of step  4  :

Step  5  :

              6x3 - 10x2  
 Simplify   ——————————————
            3x3 - 2x2 - 5x

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   6x³ - 10x²  =   2x² • (3x - 5) 

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   3x³ - 2x² - 5x  =   x • (3x² - 2x - 5) 

Trying to factor by splitting the middle term

 7.2     Factoring  3x² - 2x - 5 

The first term is,  3x²  its coefficient is  3 .
The middle term is,  -2x  its coefficient is  -2 .
The last term, "the constant", is  -5 

Step-1 : Multiply the coefficient of the first term by the constant   3 • -5 = -15 

Step-2 : Find two factors of  -15  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,  -5  and  3 
                     3x² - 5x + 3x - 5

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (3x-5)
              Add up the last 2 terms, pulling out common factors :
                     1 • (3x-5)
Step-5 : Add up the four terms of step 4 :
                    (x+1)  •  (3x-5)
             Which is the desired factorization

Canceling Out :

 7.3    Cancel out  (3x-5)  which appears on both sides of the fraction line.

Dividing exponential expressions :

 7.4    x² divided by x¹ = x^{(2 - 1)} = x¹ = x

Final result :

    2x 
  —————
  x + 1