SOLUTION: r^3+7r^2-18r factor the above x^2y-9xy^2-36y^3

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Question 117565: r^3+7r^2-18r
factor the above
x^2y-9xy^2-36y^3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
#1



r%5E3%2B7r%5E2-18r Start with the given expression


r%28r%5E2%2B7r-18%29 Factor out the GCF r


Now let's focus on the inner expression r%5E2%2B7r-18



Looking at r%5E2%2B7r-18 we can see that the first term is r%5E2 and the last term is -18 where the coefficients are 1 and -18 respectively.

Now multiply the first coefficient 1 and the last coefficient -18 to get -18. Now what two numbers multiply to -18 and add to the middle coefficient 7? Let's list all of the factors of -18:



Factors of -18:
1,2,3,6,9,18

-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -18
(1)*(-18)
(2)*(-9)
(3)*(-6)
(-1)*(18)
(-2)*(9)
(-3)*(6)

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


Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7

First NumberSecond NumberSum
1-181+(-18)=-17
2-92+(-9)=-7
3-63+(-6)=-3
-118-1+18=17
-29-2+9=7
-36-3+6=3



From this list we can see that -2 and 9 add up to 7 and multiply to -18


Now looking at the expression r%5E2%2B7r-18, replace 7r with -2r%2B9r (notice -2r%2B9r adds up to 7r. So it is equivalent to 7r)

r%5E2%2Bhighlight%28-2r%2B9r%29%2B-18


Now let's factor r%5E2-2r%2B9r-18 by grouping:


%28r%5E2-2r%29%2B%289r-18%29 Group like terms


r%28r-2%29%2B9%28r-2%29 Factor out the GCF of r out of the first group. Factor out the GCF of 9 out of the second group


%28r%2B9%29%28r-2%29 Since we have a common term of r-2, we can combine like terms



So r%5E2%2B7r-18 factors to %28r%2B9%29%28r-2%29


r%28r%2B9%29%28r-2%29 Now reintroduce the GCF r

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Answer:

So r%5E3%2B7r%5E2-18r factors to r%28r%2B9%29%28r-2%29





#2




Looking at x%5E2y-9xy%5E2-36y%5E3 we can see that the first term is x%5E2y and the last term is -36y%5E3 where the coefficients are 1 and -36 respectively.

Now multiply the first coefficient 1 and the last coefficient -36 to get -36. Now what two numbers multiply to -36 and add to the middle coefficient -9? Let's list all of the factors of -36:



Factors of -36:
1,2,3,4,6,9,12,18

-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -36
(1)*(-36)
(2)*(-18)
(3)*(-12)
(4)*(-9)
(-1)*(36)
(-2)*(18)
(-3)*(12)
(-4)*(9)

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


Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9

First NumberSecond NumberSum
1-361+(-36)=-35
2-182+(-18)=-16
3-123+(-12)=-9
4-94+(-9)=-5
-136-1+36=35
-218-2+18=16
-312-3+12=9
-49-4+9=5



From this list we can see that 3 and -12 add up to -9 and multiply to -36


Now looking at the expression x%5E2y-9xy%5E2-36y%5E3, replace -9xy%5E2 with 3xy%5E2%2B-12xy%5E2 (notice 3xy%5E2%2B-12xy%5E2 adds up to -9xy%5E2. So it is equivalent to -9xy%5E2)

x%5E2y%2Bhighlight%283xy%5E2%2B-12xy%5E2%29%2B-36y%5E3


Now let's factor x%5E2y%2B3xy%5E2-12xy%5E2-36y%5E3 by grouping:


%28x%5E2y%2B3xy%5E2%29%2B%28-12xy%5E2-36y%5E3%29 Group like terms


xy%28x%2B3y%29-12y%5E2%28x%2B3y%29 Factor out the GCF of xy out of the first group. Factor out the GCF of -12y%5E2 out of the second group


%28xy-12y%5E2%29%28x%2B3y%29 Since we have a common term of x%2B3y, we can combine like terms

So x%5E2y%2B3xy%5E2-12xy%5E2-36y%5E3 factors to %28xy-12y%5E2%29%28x%2B3y%29


So this also means that x%5E2y-9xy%5E2-36y%5E3 factors to %28xy-12y%5E2%29%28x%2B3y%29 (since x%5E2y-9xy%5E2-36y%5E3 is equivalent to x%5E2y%2B3xy%5E2-12xy%5E2-36y%5E3)

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Answer:

So x%5E2y-9xy%5E2-36y%5E3 factors to %28xy-12y%5E2%29%28x%2B3y%29