SOLUTION: i need to factor this problem: 10z^2-17z+3 can you help

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Question 244026: i need to factor this problem: 10z^2-17z+3 can you help
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 10z%5E2-17z%2B3, we can see that the first coefficient is 10, the second coefficient is -17, and the last term is 3.


Now multiply the first coefficient 10 by the last term 3 to get %2810%29%283%29=30.


Now the question is: what two whole numbers multiply to 30 (the previous product) and add to the second coefficient -17?


To find these two numbers, we need to list all of the factors of 30 (the previous product).


Factors of 30:
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 30.
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 -17:


First NumberSecond NumberSum
1301+30=31
2152+15=17
3103+10=13
565+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 -2 and -15 add to -17 (the middle coefficient).


So the two numbers -2 and -15 both multiply to 30 and add to -17


Now replace the middle term -17z with -2z-15z. Remember, -2 and -15 add to -17. So this shows us that -2z-15z=-17z.


10z%5E2%2Bhighlight%28-2z-15z%29%2B3 Replace the second term -17z with -2z-15z.


%2810z%5E2-2z%29%2B%28-15z%2B3%29 Group the terms into two pairs.


2z%285z-1%29%2B%28-15z%2B3%29 Factor out the GCF 2z from the first group.


2z%285z-1%29-3%285z-1%29 Factor out 3 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.


%282z-3%29%285z-1%29 Combine like terms. Or factor out the common term 5z-1

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Answer:


So 10z%5E2-17z%2B3 factors to %282z-3%29%285z-1%29.


Note: you can check the answer by FOILing %282z-3%29%285z-1%29 to get 10z%5E2-17z%2B3 or by graphing the original expression and the answer (the two graphs should be identical).