SOLUTION: Factor the following polynomial completely. 20x2 + 22xy + 6y2 =

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Question 307198: Factor the following polynomial completely.
20x2 + 22xy + 6y2 =

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

20x%5E2%2B22xy%2B6y%5E2 Start with the given expression


2%2810x%5E2%2B11xy%2B3y%5E2%29 Factor out the GCF 2


Now let's focus on the inner expression 10x%5E2%2B11xy%2B3y%5E2




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Looking at 10x%5E2%2B11xy%2B3y%5E2 we can see that the first term is 10x%5E2 and the last term is 3y%5E2 where the coefficients are 10 and 3 respectively.

Now multiply the first coefficient 10 and the last coefficient 3 to get 30. Now what two numbers multiply to 30 and add to the middle coefficient 11? 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 two negative numbers multiplied together make a positive number


Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11

First NumberSecond NumberSum
1301+30=31
2152+15=17
3103+10=13
565+6=11
-1-30-1+(-30)=-31
-2-15-2+(-15)=-17
-3-10-3+(-10)=-13
-5-6-5+(-6)=-11



From this list we can see that 5 and 6 add up to 11 and multiply to 30


Now looking at the expression 10x%5E2%2B11xy%2B3y%5E2, replace 11xy with 5xy%2B6xy (notice 5xy%2B6xy adds up to 11xy. So it is equivalent to 11xy)

10x%5E2%2Bhighlight%285xy%2B6xy%29%2B3y%5E2


Now let's factor 10x%5E2%2B5xy%2B6xy%2B3y%5E2 by grouping:


%2810x%5E2%2B5xy%29%2B%286xy%2B3y%5E2%29 Group like terms


5x%282x%2By%29%2B3y%282x%2By%29 Factor out the GCF of 5x out of the first group. Factor out the GCF of 3y out of the second group


%285x%2B3y%29%282x%2By%29 Since we have a common term of 2x%2By, we can combine like terms

So 10x%5E2%2B5xy%2B6xy%2B3y%5E2 factors to %285x%2B3y%29%282x%2By%29


So this also means that 10x%5E2%2B11xy%2B3y%5E2 factors to %285x%2B3y%29%282x%2By%29 (since 10x%5E2%2B11xy%2B3y%5E2 is equivalent to 10x%5E2%2B5xy%2B6xy%2B3y%5E2)



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So our expression goes from 2%2810x%5E2%2B11xy%2B3y%5E2%29 and factors further to 2%285x%2B3y%29%282x%2By%29


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Answer:

So 20x%5E2%2B22xy%2B6y%5E2 factors to 2%285x%2B3y%29%282x%2By%29

In other words, 20x%5E2%2B22xy%2B6y%5E2=2%285x%2B3y%29%282x%2By%29