SOLUTION: Can someone help me factor the following problems? 1. 36x^2 + 96xy + 64y^2 2. 8x^3 + 27y^3

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Question 317820: Can someone help me factor the following problems?
1. 36x^2 + 96xy + 64y^2
2. 8x^3 + 27y^3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1



36x%5E2%2B96xy%2B64y%5E2 Start with the given expression


4%289x%5E2%2B24xy%2B16y%5E2%29 Factor out the GCF 4


Now let's focus on the inner expression 9x%5E2%2B24xy%2B16y%5E2




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Looking at 9x%5E2%2B24xy%2B16y%5E2 we can see that the first term is 9x%5E2 and the last term is 16y%5E2 where the coefficients are 9 and 16 respectively.

Now multiply the first coefficient 9 and the last coefficient 16 to get 144. Now what two numbers multiply to 144 and add to the middle coefficient 24? Let's list all of the factors of 144:



Factors of 144:
1,2,3,4,6,8,9,12,16,18,24,36,48,72

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 144
1*144
2*72
3*48
4*36
6*24
8*18
9*16
12*12
(-1)*(-144)
(-2)*(-72)
(-3)*(-48)
(-4)*(-36)
(-6)*(-24)
(-8)*(-18)
(-9)*(-16)
(-12)*(-12)

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


Now which of these pairs add to 24? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 24

First NumberSecond NumberSum
11441+144=145
2722+72=74
3483+48=51
4364+36=40
6246+24=30
8188+18=26
9169+16=25
121212+12=24
-1-144-1+(-144)=-145
-2-72-2+(-72)=-74
-3-48-3+(-48)=-51
-4-36-4+(-36)=-40
-6-24-6+(-24)=-30
-8-18-8+(-18)=-26
-9-16-9+(-16)=-25
-12-12-12+(-12)=-24



From this list we can see that 12 and 12 add up to 24 and multiply to 144


Now looking at the expression 9x%5E2%2B24xy%2B16y%5E2, replace 24xy with 12xy%2B12xy (notice 12xy%2B12xy adds up to 24xy. So it is equivalent to 24xy)

9x%5E2%2Bhighlight%2812xy%2B12xy%29%2B16y%5E2


Now let's factor 9x%5E2%2B12xy%2B12xy%2B16y%5E2 by grouping:


%289x%5E2%2B12xy%29%2B%2812xy%2B16y%5E2%29 Group like terms


3x%283x%2B4y%29%2B4y%283x%2B4y%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of 4y out of the second group


%283x%2B4y%29%283x%2B4y%29 Since we have a common term of 3x%2B4y, we can combine like terms

So 9x%5E2%2B12xy%2B12xy%2B16y%5E2 factors to %283x%2B4y%29%283x%2B4y%29


So this also means that 9x%5E2%2B24xy%2B16y%5E2 factors to %283x%2B4y%29%283x%2B4y%29 (since 9x%5E2%2B24xy%2B16y%5E2 is equivalent to 9x%5E2%2B12xy%2B12xy%2B16y%5E2)


note: %283x%2B4y%29%283x%2B4y%29 is equivalent to %283x%2B4y%29%5E2 since the term 3x%2B4y occurs twice. So 9x%5E2%2B24xy%2B16y%5E2 also factors to %283x%2B4y%29%5E2



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So our expression goes from 4%289x%5E2%2B24xy%2B16y%5E2%29 and factors further to 4%283x%2B4y%29%5E2


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Answer:

So 36x%5E2%2B96xy%2B64y%5E2 factors to 4%283x%2B4y%29%5E2

In other words 36x%5E2%2B96xy%2B64y%5E2=4%283x%2B4y%29%5E2

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# 2



8x%5E3%2B27y%5E3 Start with the given expression.


%282x%29%5E3%2B%283y%29%5E3 Rewrite 8x%5E3 as %282x%29%5E3. Rewrite 27y%5E3 as %283y%29%5E3.


%282x%2B3y%29%28%282x%29%5E2-%282x%29%283y%29%2B%283y%29%5E2%29 Now factor by using the sum of cubes formula. Remember the sum of cubes formula is A%5E3%2BB%5E3=%28A%2BB%29%28A%5E2-AB%2BB%5E2%29


%282x%2B3y%29%284x%5E2-6xy%2B9y%5E2%29 Multiply

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Answer:

So 8x%5E3%2B27y%5E3 factors to %282x%2B3y%29%284x%5E2-6xy%2B9y%5E2%29.

In other words, 8x%5E3%2B27y%5E3=%282x%2B3y%29%284x%5E2-6xy%2B9y%5E2%29