SOLUTION: when z^2+5z-84 is factored completely, one of the factors is:

Question 131050: when z^2+5z-84 is factored completely, one of the factors is:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Looking at we can see that the first term is and the last term is where the coefficients are 1 and -84 respectively.

Now multiply the first coefficient 1 and the last coefficient -84 to get -84. Now what two numbers multiply to -84 and add to the middle coefficient 5? Let's list all of the factors of -84:

Factors of -84:
1,2,3,4,6,7,12,14,21,28,42,84

-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -84
(1)*(-84)
(2)*(-42)
(3)*(-28)
(4)*(-21)
(6)*(-14)
(7)*(-12)
(-1)*(84)
(-2)*(42)
(-3)*(28)
(-4)*(21)
(-6)*(14)
(-7)*(12)

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

Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5

First Number	Second Number	Sum
1	-84	1+(-84)=-83
2	-42	2+(-42)=-40
3	-28	3+(-28)=-25
4	-21	4+(-21)=-17
6	-14	6+(-14)=-8
7	-12	7+(-12)=-5
-1	84	-1+84=83
-2	42	-2+42=40
-3	28	-3+28=25
-4	21	-4+21=17
-6	14	-6+14=8
-7	12	-7+12=5

From this list we can see that -7 and 12 add up to 5 and multiply to -84

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

So this also means that factors to (since is equivalent to )

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Answer:
So factors to which means that one of the factors are either or
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