SOLUTION: 3a^2+12ab+12b^2 can you show me how this is done completely Factoring Trinomials #2 8x^2-26x-15 Thank you

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Question 131011: 3a^2+12ab+12b^2 can you show me how this is done completely
Factoring Trinomials

#2 8x^2-26x-15
Thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1




3a%5E2%2B12ab%2B12b%5E2 Start with the given expression


3%28a%5E2%2B4ab%2B4b%5E2%29 Factor out the GCF 3


Now let's focus on the inner expression a%5E2%2B4ab%2B4b%5E2




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Looking at 1a%5E2%2B4ab%2B4b%5E2 we can see that the first term is 1a%5E2 and the last term is 4b%5E2 where the coefficients are 1 and 4 respectively.

Now multiply the first coefficient 1 and the last coefficient 4 to get 4. Now what two numbers multiply to 4 and add to the middle coefficient 4? Let's list all of the factors of 4:



Factors of 4:
1,2

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

These factors pair up and multiply to 4
1*4
2*2
(-1)*(-4)
(-2)*(-2)

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


Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4

First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4



From this list we can see that 2 and 2 add up to 4 and multiply to 4


Now looking at the expression 1a%5E2%2B4ab%2B4b%5E2, replace 4ab with 2ab%2B2ab (notice 2ab%2B2ab adds up to 4ab. So it is equivalent to 4ab)

1a%5E2%2Bhighlight%282ab%2B2ab%29%2B4b%5E2


Now let's factor 1a%5E2%2B2ab%2B2ab%2B4b%5E2 by grouping:


%281a%5E2%2B2ab%29%2B%282ab%2B4b%5E2%29 Group like terms


a%28a%2B2b%29%2B2b%28a%2B2b%29 Factor out the GCF of a out of the first group. Factor out the GCF of 2b out of the second group


%28a%2B2b%29%28a%2B2b%29 Since we have a common term of a%2B2b, we can combine like terms

So 1a%5E2%2B2ab%2B2ab%2B4b%5E2 factors to %28a%2B2b%29%28a%2B2b%29


So this also means that 1a%5E2%2B4ab%2B4b%5E2 factors to %28a%2B2b%29%28a%2B2b%29 (since 1a%5E2%2B4ab%2B4b%5E2 is equivalent to 1a%5E2%2B2ab%2B2ab%2B4b%5E2)


note: %28a%2B2b%29%28a%2B2b%29 is equivalent to %28a%2B2b%29%5E2 since the term a%2B2b occurs twice. So 1a%5E2%2B4ab%2B4b%5E2 also factors to %28a%2B2b%29%5E2



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So our expression goes from 3%28a%5E2%2B4ab%2B4b%5E2%29 and factors further to 3%28a%2B2b%29%5E2


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Answer:

So 3a%5E2%2B12ab%2B12b%5E2 factors to 3%28a%2B2b%29%5E2








# 2




Looking at 8x%5E2-26x-15 we can see that the first term is 8x%5E2 and the last term is -15 where the coefficients are 8 and -15 respectively.

Now multiply the first coefficient 8 and the last coefficient -15 to get -120. Now what two numbers multiply to -120 and add to the middle coefficient -26? Let's list all of the factors of -120:



Factors of -120:
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120

-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -120
(1)*(-120)
(2)*(-60)
(3)*(-40)
(4)*(-30)
(5)*(-24)
(6)*(-20)
(8)*(-15)
(10)*(-12)
(-1)*(120)
(-2)*(60)
(-3)*(40)
(-4)*(30)
(-5)*(24)
(-6)*(20)
(-8)*(15)
(-10)*(12)

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


Now which of these pairs add to -26? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -26

First NumberSecond NumberSum
1-1201+(-120)=-119
2-602+(-60)=-58
3-403+(-40)=-37
4-304+(-30)=-26
5-245+(-24)=-19
6-206+(-20)=-14
8-158+(-15)=-7
10-1210+(-12)=-2
-1120-1+120=119
-260-2+60=58
-340-3+40=37
-430-4+30=26
-524-5+24=19
-620-6+20=14
-815-8+15=7
-1012-10+12=2



From this list we can see that 4 and -30 add up to -26 and multiply to -120


Now looking at the expression 8x%5E2-26x-15, replace -26x with 4x%2B-30x (notice 4x%2B-30x adds up to -26x. So it is equivalent to -26x)

8x%5E2%2Bhighlight%284x%2B-30x%29%2B-15


Now let's factor 8x%5E2%2B4x-30x-15 by grouping:


%288x%5E2%2B4x%29%2B%28-30x-15%29 Group like terms


4x%282x%2B1%29-15%282x%2B1%29 Factor out the GCF of 4x out of the first group. Factor out the GCF of -15 out of the second group


%284x-15%29%282x%2B1%29 Since we have a common term of 2x%2B1, we can combine like terms

So 8x%5E2%2B4x-30x-15 factors to %284x-15%29%282x%2B1%29


So this also means that 8x%5E2-26x-15 factors to %284x-15%29%282x%2B1%29 (since 8x%5E2-26x-15 is equivalent to 8x%5E2%2B4x-30x-15)



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Answer:
So 8x%5E2-26x-15 factors to %284x-15%29%282x%2B1%29