SOLUTION: Factoring these problems ? . 1- 18xy^3 + 3xy^2 - 10xy. 2- 15x^2 + 7x - 2. 3- 25x^2 + 20x + 4.

Algebra ->  Distributive-associative-commutative-properties -> SOLUTION: Factoring these problems ? . 1- 18xy^3 + 3xy^2 - 10xy. 2- 15x^2 + 7x - 2. 3- 25x^2 + 20x + 4.       Log On


   

Question 114003: Factoring these problems ? .
1- 18xy^3 + 3xy^2 - 10xy.
2- 15x^2 + 7x - 2.
3- 25x^2 + 20x + 4.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
#1

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 18y%5E2%2B3y-10


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 18y%5E2%2B3y-10, replace 3y with -12y%2B15y (notice -12y%2B15y adds up to 3y. So it is equivalent to 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


%286y%2B5%29%283y-2%29 Since we have a common term of 3y-2, we can combine like terms

So 18y%5E2%2B3y-10 factors to %286y%2B5%29%283y-2%29


xy%286y%2B5%29%283y-2%29 Now reintroduce the GCF



--------------------
Answer:

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





#2



Looking at 15x%5E2%2B7x-2 we can see that the first term is 15x%5E2 and the last term is -2 where the coefficients are 15 and -2 respectively.

Now multiply the first coefficient 15 and the last coefficient -2 to get -30. Now what two numbers multiply to -30 and add to the middle coefficient 7? Let's list all of the factors of -30:



Factors of -30:
1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -30
(1)*(-30)
(2)*(-15)
(3)*(-10)
(5)*(-6)
(-1)*(30)
(-2)*(15)
(-3)*(10)
(-5)*(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
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1



From this list we can see that -3 and 10 add up to 7 and multiply to -30


Now looking at the expression 15x%5E2%2B7x-2, replace 7x with -3x%2B10x (notice -3x%2B10x adds up to 7x. So it is equivalent to 7x)

15x%5E2%2Bhighlight%28-3x%2B10x%29%2B-2


Now let's factor 15x%5E2-3x%2B10x-2 by grouping:


%2815x%5E2-3x%29%2B%2810x-2%29 Group like terms


3x%285x-1%29%2B2%285x-1%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of 2 out of the second group


%283x%2B2%29%285x-1%29 Since we have a common term of 5x-1, we can combine like terms

So 15x%5E2-3x%2B10x-2 factors to %283x%2B2%29%285x-1%29


So this also means that 15x%5E2%2B7x-2 factors to %283x%2B2%29%285x-1%29 (since 15x%5E2%2B7x-2 is equivalent to 15x%5E2-3x%2B10x-2)




#3



Looking at 25x%5E2%2B20x%2B4 we can see that the first term is 25x%5E2 and the last term is 4 where the coefficients are 25 and 4 respectively.

Now multiply the first coefficient 25 and the last coefficient 4 to get 100. Now what two numbers multiply to 100 and add to the middle coefficient 20? Let's list all of the factors of 100:



Factors of 100:
1,2,4,5,10,20,25,50

-1,-2,-4,-5,-10,-20,-25,-50 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 100
1*100
2*50
4*25
5*20
10*10
(-1)*(-100)
(-2)*(-50)
(-4)*(-25)
(-5)*(-20)
(-10)*(-10)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20

First NumberSecond NumberSum
11001+100=101
2502+50=52
4254+25=29
5205+20=25
101010+10=20
-1-100-1+(-100)=-101
-2-50-2+(-50)=-52
-4-25-4+(-25)=-29
-5-20-5+(-20)=-25
-10-10-10+(-10)=-20



From this list we can see that 10 and 10 add up to 20 and multiply to 100


Now looking at the expression 25x%5E2%2B20x%2B4, replace 20x with 10x%2B10x (notice 10x%2B10x adds up to 20x. So it is equivalent to 20x)

25x%5E2%2Bhighlight%2810x%2B10x%29%2B4


Now let's factor 25x%5E2%2B10x%2B10x%2B4 by grouping:


%2825x%5E2%2B10x%29%2B%2810x%2B4%29 Group like terms


5x%285x%2B2%29%2B2%285x%2B2%29 Factor out the GCF of 5x out of the first group. Factor out the GCF of 2 out of the second group


%285x%2B2%29%285x%2B2%29 Since we have a common term of 5x%2B2, we can combine like terms

So 25x%5E2%2B10x%2B10x%2B4 factors to %285x%2B2%29%285x%2B2%29


So this also means that 25x%5E2%2B20x%2B4 factors to %285x%2B2%29%285x%2B2%29 (since 25x%5E2%2B20x%2B4 is equivalent to 25x%5E2%2B10x%2B10x%2B4)