Question

What is the quotient in simplest form state any restrictions on the variable z^2-4/z-3/z+2/z^2+z-12?

Answer 1

The expression is:


       (z^2 - 4)
       _______
      
          z - 3
 _______________

          z + 2
     ___________

       z^2 + z - 12


You can factor the numerator z^2 - 4 = (z + 2) (z - 2)

And the denominatior z^2 + z + 12 = (z + 4)(z - 3)


That permits to write the quotient as:


        (z + 2)(z - 2)
       ___________
      
             z - 3
  _______________

             z + 2
     ___________

       (z + 4)(z - 3)


Now you can multiplicate the numerator of the numerator times the denominator of the denominator, and multtiplicate the denominator of the numerator times the numerator of the denominator to obtain:


(z + 2)(z - 2)(z + 4)(z - 3)
____________________

       (z - 3)((z + 2)


Cancel the factors (z - 3) and (z + 2) because they are in both the numerator and the denominator =>

(z - 2)(z + 4) = z^2 + 2z - 8


The restrictions are that none of the cancelled factors can be 0, so z ≠ 3 and z ≠- 4.

Answer 2

Answer:

Step-by-step explanation:

The expression is:

      (z^2 - 4)

      _______

     

         z - 3

 _______________

         z + 2

    ___________

      z^2 + z - 12

You can factor the numerator z^2 - 4 = (z + 2) (z - 2)

And the denominatior z^2 + z + 12 = (z + 4)(z - 3)

That permits to write the quotient as:

       (z + 2)(z - 2)

      ___________

     

            z - 3

 _______________

            z + 2

    ___________

      (z + 4)(z - 3)