SOLUTION: my problem is factoring the trinomial completely by factoring the GCF first.........I don't know what to do.........this is my equation.. -w^2-8w+105 +-(w?)(w?) please help me..

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: my problem is factoring the trinomial completely by factoring the GCF first.........I don't know what to do.........this is my equation.. -w^2-8w+105 +-(w?)(w?) please help me..      Log On


   

Question 207387: my problem is factoring the trinomial completely by factoring the GCF first.........I don't know what to do.........this is my equation..
-w^2-8w+105
+-(w?)(w?)
please help me............thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

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-w%5E2-8w%2B105 Start with the given expression.


-%28w%5E2%2B8w-105%29 Factor out the GCF -1.


Now let's try to factor the inner expression w%5E2%2B8w-105


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Looking at the expression w%5E2%2B8w-105, we can see that the first coefficient is 1, the second coefficient is 8, and the last term is -105.


Now multiply the first coefficient 1 by the last term -105 to get %281%29%28-105%29=-105.


Now the question is: what two whole numbers multiply to -105 (the previous product) and add to the second coefficient 8?


To find these two numbers, we need to list all of the factors of -105 (the previous product).


Factors of -105:
1,3,5,7,15,21,35,105
-1,-3,-5,-7,-15,-21,-35,-105


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


These factors pair up and multiply to -105.
1*(-105) = -105
3*(-35) = -105
5*(-21) = -105
7*(-15) = -105
(-1)*(105) = -105
(-3)*(35) = -105
(-5)*(21) = -105
(-7)*(15) = -105

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 8:


First NumberSecond NumberSum
1-1051+(-105)=-104
3-353+(-35)=-32
5-215+(-21)=-16
7-157+(-15)=-8
-1105-1+105=104
-335-3+35=32
-521-5+21=16
-715-7+15=8



From the table, we can see that the two numbers -7 and 15 add to 8 (the middle coefficient).


So the two numbers -7 and 15 both multiply to -105 and add to 8


Now replace the middle term 8w with -7w%2B15w. Remember, -7 and 15 add to 8. So this shows us that -7w%2B15w=8w.


w%5E2%2Bhighlight%28-7w%2B15w%29-105 Replace the second term 8w with -7w%2B15w.


%28w%5E2-7w%29%2B%2815w-105%29 Group the terms into two pairs.


w%28w-7%29%2B%2815w-105%29 Factor out the GCF w from the first group.


w%28w-7%29%2B15%28w-7%29 Factor out 15 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.


%28w%2B15%29%28w-7%29 Combine like terms. Or factor out the common term w-7


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So -%28w%5E2%2B8w-105%29 then factors further to -%28w%2B15%29%28w-7%29


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Answer:


So -w%5E2-8w%2B105 completely factors to -%28w%2B15%29%28w-7%29.


In other words, -w%5E2-8w%2B105=-%28w%2B15%29%28w-7%29.


Note: you can check the answer by expanding -%28w%2B15%29%28w-7%29 to get -w%5E2-8w%2B105 or by graphing the original expression and the answer (the two graphs should be identical).

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