How do you multiply #(3y^2+18y+15)/(6y+6)*(y-5)/(y^2-25)#?

Question

Eva Abramson · Accepted Answer

#(3y^2 + 18y + 15)/(6y + 6) xx (y - 5)/(y^2 - 25) =color(green)( 1/2, y != -5, -1, 5)#

For problems like these, you must start by factoring everything.

#3y^2 + 18y + 5# can be factored as:

#3(y^2 + 6y + 5) = 3(y + 5)(y + 1)#

#6y + 6# can be factored as:

#6(y + 1)#

#y - 5# is as simplified as it can be.

#y^2 - 25# can be factored as #(y + 5)(y - 5)#

Putting all of this back together:

#(3(y + 5)(y + 1))/(6(y + 1)) xx (y - 5)/((y + 5)(y - 5))#

See what you can cancel out, using the property #a/a = 1#:

#(cancel((3))cancel((y + 5))cancel((y + 1)))/(cancel((6))^2cancel((y + 1))) xx (cancel(y - 5))/(cancel((y + 5))cancel((y - 5)))#

We are left with:

#1/2#

Now, before stating the final answer, we need to determine any restrictions on the variable. These will occur when the denominator equals #0#. Hence, they can be found by setting the denominator to #0# and solving for #x#.

#6y + 6 = 0" and "y^2 - 25 = 0#

#y = -1 " and "y = +-5#

Thus, #y != -1, 5, -5#

Hopefully this helps!

Abigail Allen · Answer

undefined To multiply the given expression, you can follow these steps:

1. Factorize the numerator and denominator of each fraction:
   (3y^2 + 18y + 15) = 3(y + 3)(y + 5)
   (6y + 6) = 6(y + 1)
   (y - 5) = (y - 5)
   (y^2 - 25) = (y - 5)(y + 5)

2. Cancel out any common factors between the numerators and denominators.

3. Multiply the remaining factors together.

The simplified expression after multiplying is:
(3(y + 3)(y + 5))/(6(y + 1)(y - 5)(y + 5))