SOLUTION: Factor the trinomial. b 2 - 5 b -14

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Question 110944: Factor the trinomial.
b 2 - 5 b -14

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression b%5E2-5b-14, we can see that the first coefficient is 1, the second coefficient is -5, and the last term is -14.



Now multiply the first coefficient 1 by the last term -14 to get %281%29%28-14%29=-14.



Now the question is: what two whole numbers multiply to -14 (the previous product) and add to the second coefficient -5?



To find these two numbers, we need to list all of the factors of -14 (the previous product).



Factors of -14:

1,2,7,14

-1,-2,-7,-14



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



These factors pair up and multiply to -14.

1*(-14) = -14
2*(-7) = -14
(-1)*(14) = -14
(-2)*(7) = -14


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:



First NumberSecond NumberSum
1-141+(-14)=-13
2-72+(-7)=-5
-114-1+14=13
-27-2+7=5




From the table, we can see that the two numbers 2 and -7 add to -5 (the middle coefficient).



So the two numbers 2 and -7 both multiply to -14 and add to -5



Now replace the middle term -5b with 2b-7b. Remember, 2 and -7 add to -5. So this shows us that 2b-7b=-5b.



b%5E2%2Bhighlight%282b-7b%29-14 Replace the second term -5b with 2b-7b.



%28b%5E2%2B2b%29%2B%28-7b-14%29 Group the terms into two pairs.



b%28b%2B2%29%2B%28-7b-14%29 Factor out the GCF b from the first group.



b%28b%2B2%29-7%28b%2B2%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.



%28b-7%29%28b%2B2%29 Combine like terms. Or factor out the common term b%2B2



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Answer:



So b%5E2-5%2Ab-14 factors to %28b-7%29%28b%2B2%29.



In other words, b%5E2-5%2Ab-14=%28b-7%29%28b%2B2%29.



Note: you can check the answer by expanding %28b-7%29%28b%2B2%29 to get b%5E2-5%2Ab-14 or by graphing the original expression and the answer (the two graphs should be identical).