SOLUTION: 5z^2+19z-4

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Question 742464: 5z^2+19z-4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 5z%5E2%2B19z-4, we can see that the first coefficient is 5, the second coefficient is 19, and the last term is -4.


Now multiply the first coefficient 5 by the last term -4 to get %285%29%28-4%29=-20.


Now the question is: what two whole numbers multiply to -20 (the previous product) and add to the second coefficient 19?


To find these two numbers, we need to list all of the factors of -20 (the previous product).


Factors of -20:
1,2,4,5,10,20
-1,-2,-4,-5,-10,-20


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -20.
1*(-20) = -20
2*(-10) = -20
4*(-5) = -20
(-1)*(20) = -20
(-2)*(10) = -20
(-4)*(5) = -20

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 19:


First NumberSecond NumberSum
1-201+(-20)=-19
2-102+(-10)=-8
4-54+(-5)=-1
-120-1+20=19
-210-2+10=8
-45-4+5=1



From the table, we can see that the two numbers -1 and 20 add to 19 (the middle coefficient).


So the two numbers -1 and 20 both multiply to -20 and add to 19


Now replace the middle term 19z with -z%2B20z. Remember, -1 and 20 add to 19. So this shows us that -z%2B20z=19z.


5z%5E2%2Bhighlight%28-z%2B20z%29-4 Replace the second term 19z with -z%2B20z.


%285z%5E2-z%29%2B%2820z-4%29 Group the terms into two pairs.


z%285z-1%29%2B%2820z-4%29 Factor out the GCF z from the first group.


z%285z-1%29%2B4%285z-1%29 Factor out 4 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28z%2B4%29%285z-1%29 Combine like terms. Or factor out the common term 5z-1


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Answer:


So 5z%5E2%2B19z-4 factors to %28z%2B4%29%285z-1%29.


In other words, 5z%5E2%2B19z-4=%28z%2B4%29%285z-1%29.


Note: you can check the answer by expanding %28z%2B4%29%285z-1%29 to get 5z%5E2%2B19z-4 or by graphing the original expression and the answer (the two graphs should be identical).