Question 168974: Write the coefficient of the middle term as the sum of two numbers, whose product equals the last term.
z^2-13z+30
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now the question is: what two whole numbers multiply to (the last term) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of .
Factors of :
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*30
2*15
3*10
5*6
(-1)*(-30)
(-2)*(-15)
(-3)*(-10)
(-5)*(-6)
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | 30 | 1+30=31 | | 2 | 15 | 2+15=17 | | 3 | 10 | 3+10=13 | | 5 | 6 | 5+6=11 | | -1 | -30 | -1+(-30)=-31 | | -2 | -15 | -2+(-15)=-17 | | -3 | -10 | -3+(-10)=-13 | | -5 | -6 | -5+(-6)=-11 |
From the table, we can see that the two numbers  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
So the expression breaks down into 
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