SOLUTION: Factor completely {{{32c^2+112c+98}}}

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Question 893968: Factor completely
32c%5E2%2B112c%2B98
Answer by richwmiller(17219) About Me  (Show Source):
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


32%2Ac%5E2%2B112%2Ac%2B98 Start with the given expression.



2%2816c%5E2%2B56c%2B49%29 Factor out the GCF 2.



Now let's try to factor the inner expression 16c%5E2%2B56c%2B49



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Looking at the expression 16c%5E2%2B56c%2B49, we can see that the first coefficient is 16, the second coefficient is 56, and the last term is 49.



Now multiply the first coefficient 16 by the last term 49 to get %2816%29%2849%29=784.



Now the question is: what two whole numbers multiply to 784 (the previous product) and add to the second coefficient 56?



To find these two numbers, we need to list all of the factors of 784 (the previous product).



Factors of 784:

1,2,4,7,8,14,16,28,49,56,98,112,196,392,784

-1,-2,-4,-7,-8,-14,-16,-28,-49,-56,-98,-112,-196,-392,-784



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



These factors pair up and multiply to 784.

1*784 = 784
2*392 = 784
4*196 = 784
7*112 = 784
8*98 = 784
14*56 = 784
16*49 = 784
28*28 = 784
(-1)*(-784) = 784
(-2)*(-392) = 784
(-4)*(-196) = 784
(-7)*(-112) = 784
(-8)*(-98) = 784
(-14)*(-56) = 784
(-16)*(-49) = 784
(-28)*(-28) = 784


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 56:



First NumberSecond NumberSum
17841+784=785
23922+392=394
41964+196=200
71127+112=119
8988+98=106
145614+56=70
164916+49=65
282828+28=56
-1-784-1+(-784)=-785
-2-392-2+(-392)=-394
-4-196-4+(-196)=-200
-7-112-7+(-112)=-119
-8-98-8+(-98)=-106
-14-56-14+(-56)=-70
-16-49-16+(-49)=-65
-28-28-28+(-28)=-56




From the table, we can see that the two numbers 28 and 28 add to 56 (the middle coefficient).



So the two numbers 28 and 28 both multiply to 784 and add to 56



Now replace the middle term 56c with 28c%2B28c. Remember, 28 and 28 add to 56. So this shows us that 28c%2B28c=56c.



16c%5E2%2Bhighlight%2828c%2B28c%29%2B49 Replace the second term 56c with 28c%2B28c.



%2816c%5E2%2B28c%29%2B%2828c%2B49%29 Group the terms into two pairs.



4c%284c%2B7%29%2B%2828c%2B49%29 Factor out the GCF 4c from the first group.



4c%284c%2B7%29%2B7%284c%2B7%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.



%284c%2B7%29%284c%2B7%29 Combine like terms. Or factor out the common term 4c%2B7



%284c%2B7%29%5E2 Condense the terms.



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So 2%2816c%5E2%2B56c%2B49%29 then factors further to 2%284c%2B7%29%5E2



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Answer:



So 32%2Ac%5E2%2B112%2Ac%2B98 completely factors to 2%284c%2B7%29%5E2.



In other words, 32%2Ac%5E2%2B112%2Ac%2B98=2%284c%2B7%29%5E2.



Note: you can check the answer by expanding 2%284c%2B7%29%5E2 to get 32%2Ac%5E2%2B112%2Ac%2B98 or by graphing the original expression and the answer (the two graphs should be identical).