SOLUTION: factor completely: 6(y-4)^2-(y-4)-35

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: factor completely: 6(y-4)^2-(y-4)-35      Log On


   

Question 131216: factor completely: 6(y-4)^2-(y-4)-35
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
6%28y-4%29%5E2-%28y-4%29-35 Start with the given expression


Let z=y-4. So our expression becomes

6z%5E2-z-35 Plug in z=y-4





Looking at 6z%5E2-z-35 we can see that the first term is 6z%5E2 and the last term is -35 where the coefficients are 6 and -35 respectively.

Now multiply the first coefficient 6 and the last coefficient -35 to get -210. Now what two numbers multiply to -210 and add to the middle coefficient -1? Let's list all of the factors of -210:



Factors of -210:
1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210

-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -210
(1)*(-210)
(2)*(-105)
(3)*(-70)
(5)*(-42)
(6)*(-35)
(7)*(-30)
(10)*(-21)
(14)*(-15)
(-1)*(210)
(-2)*(105)
(-3)*(70)
(-5)*(42)
(-6)*(35)
(-7)*(30)
(-10)*(21)
(-14)*(15)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1

First NumberSecond NumberSum
1-2101+(-210)=-209
2-1052+(-105)=-103
3-703+(-70)=-67
5-425+(-42)=-37
6-356+(-35)=-29
7-307+(-30)=-23
10-2110+(-21)=-11
14-1514+(-15)=-1
-1210-1+210=209
-2105-2+105=103
-370-3+70=67
-542-5+42=37
-635-6+35=29
-730-7+30=23
-1021-10+21=11
-1415-14+15=1



From this list we can see that 14 and -15 add up to -1 and multiply to -210


Now looking at the expression 6z%5E2-z-35, replace -z with 14z%2B-15z (notice 14z%2B-15z adds up to -z. So it is equivalent to -z)

6z%5E2%2Bhighlight%2814z%2B-15z%29%2B-35


Now let's factor 6z%5E2%2B14z-15z-35 by grouping:


%286z%5E2%2B14z%29%2B%28-15z-35%29 Group like terms


2z%283z%2B7%29-5%283z%2B7%29 Factor out the GCF of 2z out of the first group. Factor out the GCF of -5 out of the second group


%282z-5%29%283z%2B7%29 Since we have a common term of 3z%2B7, we can combine like terms


So 6z%5E2-z-35 factors to %282z-5%29%283z%2B7%29

%282%28y-4%29-5%29%283%28y-4%29%2B7%29 Now replace z with y-4. Remember, we let z=y-4


%282y-8-5%29%283y-12%2B7%29 Distribute


%282y-13%29%283y-5%29 Combine like terms



-------------------------------------

Answer:

So 6%28y-4%29%5E2-%28y-4%29-35 factors to %282y-13%29%283y-5%29