SOLUTION: factor the trinomial {{{c^2+6c-16}}}

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Question 893955: factor the trinomial
c%5E2%2B6c-16
Found 2 solutions by richwmiller, MathTherapy:
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
already answered today
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression c%5E2-6c-16, we can see that the first coefficient is 1, the second coefficient is -6, and the last term is -16.



Now multiply the first coefficient 1 by the last term -16 to get %281%29%28-16%29=-16.



Now the question is: what two whole numbers multiply to -16 (the previous product) and add to the second coefficient -6?



To find these two numbers, we need to list all of the factors of -16 (the previous product).



Factors of -16:

1,2,4,8,16

-1,-2,-4,-8,-16



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



These factors pair up and multiply to -16.

1*(-16) = -16
2*(-8) = -16
4*(-4) = -16
(-1)*(16) = -16
(-2)*(8) = -16
(-4)*(4) = -16


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -6:



First NumberSecond NumberSum
1-161+(-16)=-15
2-82+(-8)=-6
4-44+(-4)=0
-116-1+16=15
-28-2+8=6
-44-4+4=0




From the table, we can see that the two numbers 2 and -8 add to -6 (the middle coefficient).



So the two numbers 2 and -8 both multiply to -16 and add to -6



Now replace the middle term -6c with 2c-8c. Remember, 2 and -8 add to -6. So this shows us that 2c-8c=-6c.



c%5E2%2Bhighlight%282c-8c%29-16 Replace the second term -6c with 2c-8c.



%28c%5E2%2B2c%29%2B%28-8c-16%29 Group the terms into two pairs.



c%28c%2B2%29%2B%28-8c-16%29 Factor out the GCF c from the first group.



c%28c%2B2%29-8%28c%2B2%29 Factor out 8 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.



%28c-8%29%28c%2B2%29 Combine like terms. Or factor out the common term c%2B2



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Answer:



So c%5E2-6%2Ac-16 factors to %28c-8%29%28c%2B2%29.



In other words, c%5E2-6%2Ac-16=%28c-8%29%28c%2B2%29.



Note: you can check the answer by expanding %28c-8%29%28c%2B2%29 to get c%5E2-6%2Ac-16 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
factor the trinomial
c%5E2%2B6c-16


c%5E2+%2B+6c+-+16

c%5E2+%2B+8c+-+2c+-+16

c^2 + 8c  - 2c - 16

c(c + 8)  - 2(c + 8)

Factors: highlight_green%28%28c+-+2%29%28c+%2B+8%29%29