rational expressions?

Question

ok, once again, i don't understand what my teacher has done here.

6x^3y-24x^2y
-------------------
9x^2y^2+27xy^2


he simplified, so that we'd have:

2x-24x^2y
--------------
3y+27xy^2


but then he wrote
=2x(x-4)
-------------
3y(x+3)


i understand that he just factored, but why is it just (x-4) and (x+3). i missed the explanation.

is this correct?

5x^4y^2 -15x^3y^3
---------------------------
10x^6y^3+40x^5y^4


x^2(5x^2y^2) -3xy(sx^2y^2)
-------------------------------------
x^3y(10x^3y^2)+4x^2y^2(10x^3y^2)


=(5x^2y^2)(x^2-3xy)
---------------------------
(10x^3y^2)(x^3+4x^2y^2)


=(x^2-3xy)
--------------
2x(x^3y+4x^2y^2)

Answer 1

This is how I would do it: (It looks like you missed a few steps in your notes or something.)

6x^3y-24x^2y
-------------------
9x^2y^2+27xy^2

= (6x^2y)(x) - (6x^2y)(4)
--------------------------------
(9xy^2)(x) + (9xy^2)(3)

= (6x^2 y)(x - 4)
----------------------
(9xy^2)(x + 3)

= (3xy)(2x)(x - 4)
----------------------
(3xy)(3y)(x + 3)

The (3xy / 3xy) cancels out

= (2x)(x - 4)
-------------------
(3y)(y + 3)

Answer 2

You must have copied the second step incorrectly because
6x^3y-24x^2y
-------------------
9x^2y^2+27xy^2

is not equal to
2x-24x^2y
--------------
3y+27xy^2

To prove it, just plug in x = 1, y =1 and you will see that -1/2 does not equal -11/15.

Divide the numerator and denominator by the greatest common divisor of all the terms. i.e, divide top and bottom by 3xy. This gives

2x^2 - 8x
------------
3xy + 9y

Factoring gives

Answer:
2x ( x - 4)
------------
3y ( x + 3)

For the second question, you factored the denominator incorrectly in your second step.
The greatest common divisor for the numerator terms is 5x^3y^2. Factor this from the top. The gcd for the denominator terms is 10x^5y^3, so factor this from the bottom. This gives

5x^3 y^2 ( x - 3y)
---------------------
10x^5 y^3 (x + 4y)

Answer:

x - 3y
-------
2x^2 y (x + 4y)

Answer 3

6x^3 y - 24 x^2y/9x^2 y^2+ 27 xy^2 =
dividing by 3xy through out
2x^2 - 8x/3xy + 9y
taking 2x common in numerator and 3y in denominator
2x(x - 4)/3y(x + 3)

Answer 4

N = 6 ( x ² )( y ) (x - 4)
D = 9 ( x ) ( y ² ) (x + 3)

N / D = [ ( 2 x ) (x - 4) ] / [ (3 y ) (x + 3 ]