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

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

Let . So our expression becomes

Plug in

Looking at we can see that the first term is and the last term is 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 Number	Second Number	Sum
1	-210	1+(-210)=-209
2	-105	2+(-105)=-103
3	-70	3+(-70)=-67
5	-42	5+(-42)=-37
6	-35	6+(-35)=-29
7	-30	7+(-30)=-23
10	-21	10+(-21)=-11
14	-15	14+(-15)=-1
-1	210	-1+210=209
-2	105	-2+105=103
-3	70	-3+70=67
-5	42	-5+42=37
-6	35	-6+35=29
-7	30	-7+30=23
-10	21	-10+21=11
-14	15	-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 , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

Now replace z with . Remember, we let

Distribute

Combine like terms

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

Answer:

So factors to
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