SOLUTION: factor each polynomial completely 2u^2-36u+162

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Question 191463: factor each polynomial completely
2u^2-36u+162
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
2u%5E2-36u%2B162 Start with the given expression


2%28u%5E2-18u%2B81%29 Factor out the GCF 2


Now let's focus on the inner expression u%5E2-18u%2B81




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Looking at u%5E2-18u%2B81 we can see that the first term is u%5E2 and the last term is 81 where the coefficients are 1 and 81 respectively.

Now multiply the first coefficient 1 and the last coefficient 81 to get 81. Now what two numbers multiply to 81 and add to the middle coefficient -18? Let's list all of the factors of 81:



Factors of 81:
1,3,9,27

-1,-3,-9,-27 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 81
1*81
3*27
9*9
(-1)*(-81)
(-3)*(-27)
(-9)*(-9)

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


Now which of these pairs add to -18? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -18

First NumberSecond NumberSum
1811+81=82
3273+27=30
999+9=18
-1-81-1+(-81)=-82
-3-27-3+(-27)=-30
-9-9-9+(-9)=-18



From this list we can see that -9 and -9 add up to -18 and multiply to 81


Now looking at the expression u%5E2-18u%2B81, replace -18u with -9u-9u (notice -9u-9u combines to -18u. So it is equivalent to -18u)

u%5E2%2Bhighlight%28-9u-9u%29%2B81


Now let's factor u%5E2-9u-9u%2B81 by grouping:


%28u%5E2-9u%29%2B%28-9u%2B81%29 Group like terms


u%28u-9%29-9%28u-9%29 Factor out the GCF of u out of the first group. Factor out the GCF of -9 out of the second group


%28u-9%29%28u-9%29 Since we have a common term of u-9, we can combine like terms

So u%5E2-9u-9u%2B81 factors to %28u-9%29%28u-9%29


So this also means that u%5E2-18u%2B81 factors to %28u-9%29%28u-9%29 (since u%5E2-18u%2B81 is equivalent to u%5E2-9u-9u%2B81)


note: %28u-9%29%28u-9%29 is equivalent to %28u-9%29%5E2 since the term u-9 occurs twice. So u%5E2-18u%2B81 also factors to %28u-9%29%5E2



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So our expression goes from 2%28u%5E2-18u%2B81%29 and factors further to 2%28u-9%29%5E2


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Answer:

So 2u%5E2-36u%2B162 completely factors to 2%28u-9%29%5E2