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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