SOLUTION: Factor the trinomial 3x^2+27x+42

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Question 190250: Factor the trinomial
3x^2+27x+42
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

3x%5E2%2B27x%2B42 Start with the given expression


3%28x%5E2%2B9x%2B14%29 Factor out the GCF 3


Now let's focus on the inner expression x%5E2%2B9x%2B14




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Looking at the expression x%5E2%2B9x%2B14, we can see that the first coefficient is 1, the second coefficient is 9, and the last term is 14.


Now multiply the first coefficient 1 by the last term 14 to get %281%29%2814%29=14.


Now the question is: what two whole numbers multiply to 14 (the previous product) and add to the second coefficient 9?


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 9:


First NumberSecond NumberSum
1141+14=15
272+7=9
-1-14-1+(-14)=-15
-2-7-2+(-7)=-9



From the table, we can see that the two numbers 2 and 7 add to 9 (the middle coefficient).


So the two numbers 2 and 7 both multiply to 14 and add to 9


Now replace the middle term 9x with 2x%2B7x. Remember, 2 and 7 add to 9. So this shows us that 2x%2B7x=9x.


x%5E2%2Bhighlight%282x%2B7x%29%2B14 Replace the second term 9x with 2x%2B7x.


%28x%5E2%2B2x%29%2B%287x%2B14%29 Group the terms into two pairs.


x%28x%2B2%29%2B%287x%2B14%29 Factor out the GCF x from the first group.


x%28x%2B2%29%2B7%28x%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.


%28x%2B7%29%28x%2B2%29 Combine like terms. Or factor out the common term x%2B2


So x%5E2%2B9x%2B14 factors to %28x%2B7%29%28x%2B2%29.




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So our expression goes from 3%28x%5E2%2B9x%2B14%29 and factors further to 3%28x%2B7%29%28x%2B2%29


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Answer:

So 3x%5E2%2B27x%2B42 completely factors to 3%28x%2B7%29%28x%2B2%29