SOLUTION: factor completely: 6y^3+11y^2-35y

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Question 131051: factor completely: 6y^3+11y^2-35y
Found 2 solutions by jim_thompson5910, Earlsdon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

6y%5E3%2B11y%5E2-35y Start with the given expression


y%286y%5E2%2B11y-35%29 Factor out the GCF y


Now let's focus on the inner expression 6y%5E2%2B11y-35




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Looking at 6y%5E2%2B11y-35 we can see that the first term is 6y%5E2 and the last term is -35 where the coefficients are 6 and -35 respectively.

Now multiply the first coefficient 6 and the last coefficient -35 to get -210. Now what two numbers multiply to -210 and add to the middle coefficient 11? Let's list all of the factors of -210:



Factors of -210:
1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210

-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -210
(1)*(-210)
(2)*(-105)
(3)*(-70)
(5)*(-42)
(6)*(-35)
(7)*(-30)
(10)*(-21)
(14)*(-15)
(-1)*(210)
(-2)*(105)
(-3)*(70)
(-5)*(42)
(-6)*(35)
(-7)*(30)
(-10)*(21)
(-14)*(15)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11

First NumberSecond NumberSum
1-2101+(-210)=-209
2-1052+(-105)=-103
3-703+(-70)=-67
5-425+(-42)=-37
6-356+(-35)=-29
7-307+(-30)=-23
10-2110+(-21)=-11
14-1514+(-15)=-1
-1210-1+210=209
-2105-2+105=103
-370-3+70=67
-542-5+42=37
-635-6+35=29
-730-7+30=23
-1021-10+21=11
-1415-14+15=1



From this list we can see that -10 and 21 add up to 11 and multiply to -210


Now looking at the expression 6y%5E2%2B11y-35, replace 11y with -10y%2B21y (notice -10y%2B21y adds up to 11y. So it is equivalent to 11y)

6y%5E2%2Bhighlight%28-10y%2B21y%29%2B-35


Now let's factor 6y%5E2-10y%2B21y-35 by grouping:


%286y%5E2-10y%29%2B%2821y-35%29 Group like terms


2y%283y-5%29%2B7%283y-5%29 Factor out the GCF of 2y out of the first group. Factor out the GCF of 7 out of the second group


%282y%2B7%29%283y-5%29 Since we have a common term of 3y-5, we can combine like terms

So 6y%5E2-10y%2B21y-35 factors to %282y%2B7%29%283y-5%29


So this also means that 6y%5E2%2B11y-35 factors to %282y%2B7%29%283y-5%29 (since 6y%5E2%2B11y-35 is equivalent to 6y%5E2-10y%2B21y-35)



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So our expression goes from y%286y%5E2%2B11y-35%29 and factors further to y%282y%2B7%29%283y-5%29


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Answer:

So 6y%5E3%2B11y%5E2-35y factors to y%282y%2B7%29%283y-5%29

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Factor:
6y%5E3%2B11y%5E2-35y First, factor a y.
y%286y%5E2%2B11y-35%29 Now factor the trinomial.
y%282y%2B7%29%283y-5%29