SOLUTION: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive. 3c^2 + 6c - 105

Algebra ->  Expressions-with-variables -> SOLUTION: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive. 3c^2 + 6c - 105       Log On


   

Question 318015: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive.
3c^2 + 6c - 105
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

3c%5E2%2B6c-105 Start with the given expression.


3%28c%5E2%2B2c-35%29 Factor out the GCF 3.


Now let's try to factor the inner expression c%5E2%2B2c-35


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Looking at the expression c%5E2%2B2c-35, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is -35.


Now multiply the first coefficient 1 by the last term -35 to get %281%29%28-35%29=-35.


Now the question is: what two whole numbers multiply to -35 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of -35 (the previous product).


Factors of -35:
1,5,7,35
-1,-5,-7,-35


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


These factors pair up and multiply to -35.
1*(-35) = -35
5*(-7) = -35
(-1)*(35) = -35
(-5)*(7) = -35

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:


First NumberSecond NumberSum
1-351+(-35)=-34
5-75+(-7)=-2
-135-1+35=34
-57-5+7=2



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


So the two numbers -5 and 7 both multiply to -35 and add to 2


Now replace the middle term 2c with -5c%2B7c. Remember, -5 and 7 add to 2. So this shows us that -5c%2B7c=2c.


c%5E2%2Bhighlight%28-5c%2B7c%29-35 Replace the second term 2c with -5c%2B7c.


%28c%5E2-5c%29%2B%287c-35%29 Group the terms into two pairs.


c%28c-5%29%2B%287c-35%29 Factor out the GCF c from the first group.


c%28c-5%29%2B7%28c-5%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.


%28c%2B7%29%28c-5%29 Combine like terms. Or factor out the common term c-5


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So 3%28c%5E2%2B2c-35%29 then factors further to 3%28c%2B7%29%28c-5%29


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Answer:


So 3c%5E2%2B6c-105 completely factors to 3%28c%2B7%29%28c-5%29.


In other words, 3c%5E2%2B6c-105=3%28c%2B7%29%28c-5%29.


Note: you can check the answer by expanding 3%28c%2B7%29%28c-5%29 to get 3c%5E2%2B6c-105 or by graphing the original expression and the answer (the two graphs should be identical).