SOLUTION: factor completely 12x^2+32x+21

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Question 167670: factor completely
12x^2+32x+21
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 12x%5E2%2B32x%2B21, we can see that the first coefficient is 12, the second coefficient is 32, and the last term is 21.


Now multiply the first coefficient 12 by the last term 21 to get %2812%29%2821%29=252.


Now the question is: what two whole numbers multiply to 252 (the previous product) and add to the second coefficient 32?


To find these two numbers, we need to list all of the factors of 252 (the previous product).


Factors of 252:
1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252
-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-63,-84,-126,-252


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


These factors pair up and multiply to 252.
1*252
2*126
3*84
4*63
6*42
7*36
9*28
12*21
14*18
(-1)*(-252)
(-2)*(-126)
(-3)*(-84)
(-4)*(-63)
(-6)*(-42)
(-7)*(-36)
(-9)*(-28)
(-12)*(-21)
(-14)*(-18)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 32:


First NumberSecond NumberSum
12521+252=253
21262+126=128
3843+84=87
4634+63=67
6426+42=48
7367+36=43
9289+28=37
122112+21=33
141814+18=32
-1-252-1+(-252)=-253
-2-126-2+(-126)=-128
-3-84-3+(-84)=-87
-4-63-4+(-63)=-67
-6-42-6+(-42)=-48
-7-36-7+(-36)=-43
-9-28-9+(-28)=-37
-12-21-12+(-21)=-33
-14-18-14+(-18)=-32



From the table, we can see that the two numbers 14 and 18 add to 32 (the middle coefficient).


So the two numbers 14 and 18 both multiply to 252 and add to 32


Now replace the middle term 32x with 14x%2B18x. Remember, 14 and 18 add to 32. So this shows us that 14x%2B18x=32x.


12x%5E2%2Bhighlight%2814x%2B18x%29%2B21 Replace the second term 32x with 14x%2B18x.


%2812x%5E2%2B14x%29%2B%2818x%2B21%29 Group the terms into two pairs.


2x%286x%2B7%29%2B%2818x%2B21%29 Factor out the GCF 2x from the first group.


2x%286x%2B7%29%2B3%286x%2B7%29 Factor out 3 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.


%282x%2B3%29%286x%2B7%29 Combine like terms. Or factor out the common term 6x%2B7

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Answer:


So 12x%5E2%2B32x%2B21 factors to %282x%2B3%29%286x%2B7%29.


Note: you can check the answer by FOILing %282x%2B3%29%286x%2B7%29 to get 12x%5E2%2B32x%2B21 or by graphing the original expression and the answer (the two graphs should be identical).