SOLUTION: 7. Ok this is impossibly hard for me. How do I complete this using factoring? Thank you so much. 18xy^3 + 3xy^2 - 10xy

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 7. Ok this is impossibly hard for me. How do I complete this using factoring? Thank you so much. 18xy^3 + 3xy^2 - 10xy      Log On


   

Question 156341: 7. Ok this is impossibly hard for me. How do I complete this using factoring? Thank you so much.


18xy^3 + 3xy^2 - 10xy
Found 2 solutions by oscargut, jim_thompson5910:
Answer by oscargut(2103) About Me  (Show Source):
You can put this solution on YOUR website!
18xy%5E3+%2B+3xy%5E2+-+10xy=xy%2818y%5E2%2B3y-10%29
18y^2+3y-10=0
then
y+=+%28-3+%2B-+sqrt%28+3%5E2-4%2A18%2A%28-10%29+%29%29%2F%282%2A18%29+ =
y+=+%28-3+%2B-+sqrt%28+3%5E2-4%2A18%2A%28-10%29+%29%29%2F%282%2A18%29+ =
y+=+%28-3+%2B-+27%29%2F%2836%29+
so roots are -30/36 and 24/36
then roots are -5/6 and 2/3
so 18xy%5E3+%2B+3xy%5E2+-+10xy+=+xy%28x%2B5%2F6%29%28x-2%2F3%29

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
18xy%5E3%2B3xy%5E2-10xy Start with the given expression


xy%2818y%5E2%2B3y-10%29 Factor out the GCF xy


Now let's focus on the inner expression 18xy%5E2%2B3y-10


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Looking at 18y%5E2%2B3y-10 we can see that the first term is 18y%5E2 and the last term is -10 where the coefficients are 18 and -10 respectively.

Now multiply the first coefficient 18 and the last coefficient -10 to get -180. Now what two numbers multiply to -180 and add to the middle coefficient 3? Let's list all of the factors of -180:



Factors of -180:
1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180

-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-30,-36,-45,-60,-90,-180 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -180
(1)*(-180)
(2)*(-90)
(3)*(-60)
(4)*(-45)
(5)*(-36)
(6)*(-30)
(9)*(-20)
(10)*(-18)
(12)*(-15)
(-1)*(180)
(-2)*(90)
(-3)*(60)
(-4)*(45)
(-5)*(36)
(-6)*(30)
(-9)*(20)
(-10)*(18)
(-12)*(15)

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


Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3

First NumberSecond NumberSum
1-1801+(-180)=-179
2-902+(-90)=-88
3-603+(-60)=-57
4-454+(-45)=-41
5-365+(-36)=-31
6-306+(-30)=-24
9-209+(-20)=-11
10-1810+(-18)=-8
12-1512+(-15)=-3
-1180-1+180=179
-290-2+90=88
-360-3+60=57
-445-4+45=41
-536-5+36=31
-630-6+30=24
-920-9+20=11
-1018-10+18=8
-1215-12+15=3



From this list we can see that -12 and 15 add up to 3 and multiply to -180


Now looking at the expression 18xy%5E2%2B3y-10, replace 3y with -12y%2B15y (Remember, if -12%2B15=3, then -12y%2B15y=3y)


18y%5E2%2Bhighlight%28-12y%2B15y%29%2B-10


Now let's factor 18y%5E2-12y%2B15y-10 by grouping:


%2818y%5E2-12y%29%2B%2815y-10%29 Group like terms


6y%283y-2%29%2B5%283y-2%29 Factor out the GCF of 6y out of the first group. Factor out the GCF of 5 out of the second group


So this also means that 18y%5E2%2B3y-10 factors to %286y%2B5%29%283y-2%29 (since 18y%5E2%2B3y-10 is equivalent to 18y%5E2-12y%2B15y-10)



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So the expression xy%2818y%5E2%2B3y-10%29 factors further to xy%286y%2B5%29%283y-2%29



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Answer:


So 18xy%5E3%2B3xy%5E2-10xy completely factors to xy%286y%2B5%29%283y-2%29


To check the answer, simply FOIL and expand xy%286y%2B5%29%283y-2%29 to get 18xy%5E3%2B3xy%5E2-10xy