SOLUTION: Factoring help. 30v^2+186v-432

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Question 253128: Factoring help.
30v^2+186v-432
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

30v%5E2%2B186v-432 Start with the given expression.


6%285v%5E2%2B31v-72%29 Factor out the GCF 6.


Now let's try to factor the inner expression 5v%5E2%2B31v-72


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Looking at the expression 5v%5E2%2B31v-72, we can see that the first coefficient is 5, the second coefficient is 31, and the last term is -72.


Now multiply the first coefficient 5 by the last term -72 to get %285%29%28-72%29=-360.


Now the question is: what two whole numbers multiply to -360 (the previous product) and add to the second coefficient 31?


To find these two numbers, we need to list all of the factors of -360 (the previous product).


Factors of -360:
1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360
-1,-2,-3,-4,-5,-6,-8,-9,-10,-12,-15,-18,-20,-24,-30,-36,-40,-45,-60,-72,-90,-120,-180,-360


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


These factors pair up and multiply to -360.
1*(-360) = -360
2*(-180) = -360
3*(-120) = -360
4*(-90) = -360
5*(-72) = -360
6*(-60) = -360
8*(-45) = -360
9*(-40) = -360
10*(-36) = -360
12*(-30) = -360
15*(-24) = -360
18*(-20) = -360
(-1)*(360) = -360
(-2)*(180) = -360
(-3)*(120) = -360
(-4)*(90) = -360
(-5)*(72) = -360
(-6)*(60) = -360
(-8)*(45) = -360
(-9)*(40) = -360
(-10)*(36) = -360
(-12)*(30) = -360
(-15)*(24) = -360
(-18)*(20) = -360

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 31:


First NumberSecond NumberSum
1-3601+(-360)=-359
2-1802+(-180)=-178
3-1203+(-120)=-117
4-904+(-90)=-86
5-725+(-72)=-67
6-606+(-60)=-54
8-458+(-45)=-37
9-409+(-40)=-31
10-3610+(-36)=-26
12-3012+(-30)=-18
15-2415+(-24)=-9
18-2018+(-20)=-2
-1360-1+360=359
-2180-2+180=178
-3120-3+120=117
-490-4+90=86
-572-5+72=67
-660-6+60=54
-845-8+45=37
-940-9+40=31
-1036-10+36=26
-1230-12+30=18
-1524-15+24=9
-1820-18+20=2



From the table, we can see that the two numbers -9 and 40 add to 31 (the middle coefficient).


So the two numbers -9 and 40 both multiply to -360 and add to 31


Now replace the middle term 31v with -9v%2B40v. Remember, -9 and 40 add to 31. So this shows us that -9v%2B40v=31v.


5v%5E2%2Bhighlight%28-9v%2B40v%29-72 Replace the second term 31v with -9v%2B40v.


%285v%5E2-9v%29%2B%2840v-72%29 Group the terms into two pairs.


v%285v-9%29%2B%2840v-72%29 Factor out the GCF v from the first group.


v%285v-9%29%2B8%285v-9%29 Factor out 8 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.


%28v%2B8%29%285v-9%29 Combine like terms. Or factor out the common term 5v-9


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So 6%285v%5E2%2B31v-72%29 then factors further to 6%28v%2B8%29%285v-9%29


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Answer:


So 30v%5E2%2B186v-432 completely factors to 6%28v%2B8%29%285v-9%29.


In other words, 30v%5E2%2B186v-432=6%28v%2B8%29%285v-9%29.


Note: you can check the answer by expanding 6%28v%2B8%29%285v-9%29 to get 30v%5E2%2B186v-432 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Factoring help.
30v^2+186v-432
---
= 3(10v^2 + 62v - 144)
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= 3*2(5v^2 + 31v - 72)
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= 6(5v^2+40v-9v-72
---
= 6(5v(v+8)-9(v+8))
= 6(v+8)(5v-9)
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Cheers,
Stan H.