SOLUTION: Factor by grouping 14b^2-55b+21

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Question 297591: Factor by grouping
14b^2-55b+21

Answer by jim_thompson5910(35256) About Me  (Show Source):
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 14b%5E2-55b%2B21, we can see that the first coefficient is 14, the second coefficient is -55, and the last term is 21.



Now multiply the first coefficient 14 by the last term 21 to get %2814%29%2821%29=294.



Now the question is: what two whole numbers multiply to 294 (the previous product) and add to the second coefficient -55?



To find these two numbers, we need to list all of the factors of 294 (the previous product).



Factors of 294:

1,2,3,6,7,14,21,42,49,98,147,294

-1,-2,-3,-6,-7,-14,-21,-42,-49,-98,-147,-294



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



These factors pair up and multiply to 294.

1*294 = 294
2*147 = 294
3*98 = 294
6*49 = 294
7*42 = 294
14*21 = 294
(-1)*(-294) = 294
(-2)*(-147) = 294
(-3)*(-98) = 294
(-6)*(-49) = 294
(-7)*(-42) = 294
(-14)*(-21) = 294


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -55:



First NumberSecond NumberSum
12941+294=295
21472+147=149
3983+98=101
6496+49=55
7427+42=49
142114+21=35
-1-294-1+(-294)=-295
-2-147-2+(-147)=-149
-3-98-3+(-98)=-101
-6-49-6+(-49)=-55
-7-42-7+(-42)=-49
-14-21-14+(-21)=-35




From the table, we can see that the two numbers -6 and -49 add to -55 (the middle coefficient).



So the two numbers -6 and -49 both multiply to 294 and add to -55



Now replace the middle term -55b with -6b-49b. Remember, -6 and -49 add to -55. So this shows us that -6b-49b=-55b.



14b%5E2%2Bhighlight%28-6b-49b%29%2B21 Replace the second term -55b with -6b-49b.



%2814b%5E2-6b%29%2B%28-49b%2B21%29 Group the terms into two pairs.



2b%287b-3%29%2B%28-49b%2B21%29 Factor out the GCF 2b from the first group.



2b%287b-3%29-7%287b-3%29 Factor out 7 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.



%282b-7%29%287b-3%29 Combine like terms. Or factor out the common term 7b-3



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Answer:



So 14%2Ab%5E2-55%2Ab%2B21 factors to %282b-7%29%287b-3%29.



In other words, 14%2Ab%5E2-55%2Ab%2B21=%282b-7%29%287b-3%29.



Note: you can check the answer by expanding %282b-7%29%287b-3%29 to get 14%2Ab%5E2-55%2Ab%2B21 or by graphing the original expression and the answer (the two graphs should be identical).