could someone help me understand this?

Question

(16z²-16z+4)/12z²-2z-2) • (12z²+19z+5)/(4-2z-12z²)

Answer 1

16z² - 16z + 4      12z² + 19z + 5

----------------- • -------------------

12z² -2z -2            4 - 2z - 12z² 

 

 

16z² - 16z + 4     12z² + 19z + 5
----------------- • -------------------
12z² -2z -2          -12z² - 2z + 4       I rearranged the denominator in the second fraction just because I like them all to "look" the same way

 

 

Now I begin by looking at each numerator and denominator to see if each of their terms have a common multiple.

 

16z² - 16z + 4  each term is a multiple of 4 so I will factor that out... 4(4z² - 4z + 1)

12z² -2z - 2  each term is a multiple of 2 so I will factor that out... 2(6z² -z-1)

12z² + 19z + 5  each term has no common multiple so it stays the same

-12z² - 2z + 4  each term is a multiple of 2 so I will factor that out... 2(-6z² - z + 2)

 

 4(4z² - 4z + 1)        12z2 + 19z + 5

-------------------  •  -------------------

2(6z² -z-1)               2(-6z² - z + 2)     Now look at the multiples we pulled out and see if you can reduce them.

 

4 over the 2 in the first fraction gives us 2 in the numerator. And because we have a 2 in the denominator of the second fraction, that allows us to reduce again, 2 over 2 giving us 1. So technically, all the multiples we pulled out reduce to 1 and we don't have to actually write a 1 b/c anything multiplied by 1 is itself.

 

Now we're left with....

 

4z² - 4z + 1        12z² + 19z + 5

---------------  •  -------------------

6z² -z-1               -(6z² + z - 2)      

 

Note: I factored out a -1 to make it easier to deal with a positive coefficient on the 6z² when factoring.

 

Let's factor each numerator and denominator...

 

(2z - 1)(2z - 1)          (4z + 5)(3z + 1)

------------------   •   --------------------

(3z + 1)(2z - 1)        -(3z + 2)(2z - 1)

 

Look for any like terms we can reduce to 1. For example, (2z - 1) / (2z - 1) reduces to 1 and again, we do not have to write the 1. When all like terms are reduced to 1, we are left with this fraction...

 

 (4z + 5)

-----------

-(3z + 2)

 

And that is your answer!