Question

Simplify x^2+y^2-z^2+2xy/x^2-y^2-z^2-2yz

Answer 1

answer:

x3 - 2xz2 - xy2 - 2y

 ————————————————————

          x          

Step-by-step explanation:

x2-z2+2yz-2xy/x2-y2-2yz-z2

Final result :

 x3 - 2xz2 - xy2 - 2y

 ————————————————————

          x          

Step by step solution :

Step  1  :

            y

Simplify   ——

           x2

Equation at the end of step  1  :

                           y

 ((((((x2)-(z2))+2zy)-(2x•——))-y2)-2zy)-z2

                          x2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  x  as the denominator :

                     x2 - z2 + 2zy     (x2 - z2 + 2zy) • x

    x2 - z2 + 2zy =  —————————————  =  ———————————————————

                           1                    x        

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

Trying to factor a multi variable polynomial :

2.2    Factoring    x2 - z2 + 2zy

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

Factorization fails

Adding fractions that have a common denominator :

2.3       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:

(x2-z2+2zy) • x - (2y)     x3 - xz2 + 2xzy - 2y

——————————————————————  =  ————————————————————

          x                         x          

Equation at the end of step  2  :

   (x3 - xz2 + 2xzy - 2y)                    

 ((—————————————————————— -  y2) -  2zy) -  z2

             x                              

Step  3  :

Rewriting the whole as an Equivalent Fraction :

3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x  as the denominator :

         y2     y2 • x

   y2 =  ——  =  ——————

         1        x  

Checking for a perfect cube :

3.2    x3 - xz2 + 2xzy - 2y  is not a perfect cube

Adding fractions that have a common denominator :

3.3       Adding up the two equivalent fractions

(x3-xz2+2xzy-2y) - (y2 • x)      x3 - xz2 + 2xzy - xy2 - 2y

———————————————————————————  =  ——————————————————————————

             x                              x            

Equation at the end of step  3  :

  (x3 - xz2 + 2xzy - xy2 - 2y)            

 (———————————————————————————— -  2zy) -  z2

               x                          

Step  4  :

Rewriting the whole as an Equivalent Fraction :

4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x  as the denominator :

          2zy     2zy • x

   2zy =  ———  =  ———————

           1         x  

Adding fractions that have a common denominator :

4.2       Adding up the two equivalent fractions

(x3-xz2+2xzy-xy2-2y) - (2zy • x)      x3 - xz2 - xy2 - 2y

————————————————————————————————  =  ———————————————————

               x                              x        

Equation at the end of step  4  :

 (x3 - xz2 - xy2 - 2y)    

 ————————————————————— -  z2

           x              

Step  5  :

Rewriting the whole as an Equivalent Fraction :

5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x  as the denominator :

         z2     z2 • x

   z2 =  ——  =  ——————

         1        x  

Checking for a perfect cube :

5.2    x3 - xz2 - xy2 - 2y  is not a perfect cube

Adding fractions that have a common denominator :

5.3       Adding up the two equivalent fractions

(x3-xz2-xy2-2y) - (z2 • x)      x3 - 2xz2 - xy2 - 2y

——————————————————————————  =  ————————————————————

            x                           x          

Checking for a perfect cube :

5.4    x3 - 2xz2 - xy2 - 2y  is not a perfect cube

Final result :

 x3 - 2xz2 - xy2 - 2y

 ————————————————————

          x