SOLUTION: Factor completely. 3(x-2)^2-3(x-2)-6

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Question 103930: Factor completely.
3(x-2)^2-3(x-2)-6
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let y=x-2

So we get

3y%5E2-3y-6


Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


3%2Ay%5E2-3%2Ay-6 Start with the given expression.



3%28y%5E2-y-2%29 Factor out the GCF 3.



Now let's try to factor the inner expression y%5E2-y-2



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Looking at the expression y%5E2-y-2, we can see that the first coefficient is 1, the second coefficient is -1, and the last term is -2.



Now multiply the first coefficient 1 by the last term -2 to get %281%29%28-2%29=-2.



Now the question is: what two whole numbers multiply to -2 (the previous product) and add to the second coefficient -1?



To find these two numbers, we need to list all of the factors of -2 (the previous product).



Factors of -2:

1,2

-1,-2



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to -2.

1*(-2) = -2
(-1)*(2) = -2


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -1:



First NumberSecond NumberSum
1-21+(-2)=-1
-12-1+2=1




From the table, we can see that the two numbers 1 and -2 add to -1 (the middle coefficient).



So the two numbers 1 and -2 both multiply to -2 and add to -1



Now replace the middle term -1y with y-2y. Remember, 1 and -2 add to -1. So this shows us that y-2y=-1y.



y%5E2%2Bhighlight%28y-2y%29-2 Replace the second term -1y with y-2y.



%28y%5E2%2By%29%2B%28-2y-2%29 Group the terms into two pairs.



y%28y%2B1%29%2B%28-2y-2%29 Factor out the GCF y from the first group.



y%28y%2B1%29-2%28y%2B1%29 Factor out 2 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%28y-2%29%28y%2B1%29 Combine like terms. Or factor out the common term y%2B1



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So 3%28y%5E2-y-2%29 then factors further to 3%28y-2%29%28y%2B1%29



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Answer:



So 3%2Ay%5E2-3%2Ay-6 completely factors to 3%28y-2%29%28y%2B1%29.



In other words, 3%2Ay%5E2-3%2Ay-6=3%28y-2%29%28y%2B1%29.



Note: you can check the answer by expanding 3%28y-2%29%28y%2B1%29 to get 3%2Ay%5E2-3%2Ay-6 or by graphing the original expression and the answer (the two graphs should be identical).




Since 3y%5E2-3y-6 factors to %283y-6%29%28y%2B1%29 we can replace y with y=x-2 to get


%283%28x%2B2%29-6%29%28%28x%2B2%29%2B1%29