SOLUTION: 18a^4b^2-24a^3b^3+8a^2b^4

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Question 168158: 18a^4b^2-24a^3b^3+8a^2b^4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

18a%5E4b%5E2-24a%5E3b%5E3%2B8a%5E2b%5E4 Start with the given expression


2a%5E2b%5E2%289a%5E2-12ab%2B4b%5E2%29 Factor out the GCF 2a%5E2b%5E2


Now let's focus on the inner expression 9a%5E2-12ab%2B4b%5E2




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Looking at 9a%5E2-12ab%2B4b%5E2 we can see that the first term is 9a%5E2 and the last term is 4b%5E2 where the coefficients are 9 and 4 respectively.

Now multiply the first coefficient 9 and the last coefficient 4 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient -12? 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
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)

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


Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12

First NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+6=12
-1-36-1+(-36)=-37
-2-18-2+(-18)=-20
-3-12-3+(-12)=-15
-4-9-4+(-9)=-13
-6-6-6+(-6)=-12



From this list we can see that -6 and -6 add up to -12 and multiply to 36


Now looking at the expression 9a%5E2-12ab%2B4b%5E2, replace -12ab with -6ab%2B-6ab (notice -6ab%2B-6ab adds up to -12ab. So it is equivalent to -12ab)

9a%5E2%2Bhighlight%28-6ab%2B-6ab%29%2B4b%5E2


Now let's factor 9a%5E2-6ab-6ab%2B4b%5E2 by grouping:


%289a%5E2-6ab%29%2B%28-6ab%2B4b%5E2%29 Group like terms


3a%283a-2b%29-2b%283a-2b%29 Factor out the GCF of 3a out of the first group. Factor out the GCF of -2b out of the second group


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

So 9a%5E2-6ab-6ab%2B4b%5E2 factors to %283a-2b%29%283a-2b%29


So this also means that 9a%5E2-12ab%2B4b%5E2 factors to %283a-2b%29%283a-2b%29 (since 9a%5E2-12ab%2B4b%5E2 is equivalent to 9a%5E2-6ab-6ab%2B4b%5E2)


note: %283a-2b%29%283a-2b%29 is equivalent to %283a-2b%29%5E2 since the term 3a-2b occurs twice. So 9a%5E2-12ab%2B4b%5E2 also factors to %283a-2b%29%5E2



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So our expression goes from 2a%5E2b%5E2%289a%5E2-12ab%2B4b%5E2%29 and factors further to 2a%5E2b%5E2%283a-2b%29%5E2


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Answer:

So 18a%5E4b%5E2-24a%5E3b%5E3%2B8a%5E2b%5E4 factors to 2a%5E2b%5E2%283a-2b%29%5E2