SOLUTION: What are the steps to factoring a trinomial that has a coefficient in the first term? This is the problem that I've been having trouble with{{{2x^2+x-3}}}

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Question 133456: What are the steps to factoring a trinomial that has a coefficient in the first term? This is the problem that I've been having trouble with2x%5E2%2Bx-3
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 2x%5E2%2Bx-3, we can see that the first coefficient is 2, the second coefficient is 1, and the last term is -3.



Now multiply the first coefficient 2 by the last term -3 to get %282%29%28-3%29=-6.



Now the question is: what two whole numbers multiply to -6 (the previous product) and add to the second coefficient 1?



To find these two numbers, we need to list all of the factors of -6 (the previous product).



Factors of -6:

1,2,3,6

-1,-2,-3,-6



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



These factors pair up and multiply to -6.

1*(-6) = -6
2*(-3) = -6
(-1)*(6) = -6
(-2)*(3) = -6


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:



First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=1




From the table, we can see that the two numbers -2 and 3 add to 1 (the middle coefficient).



So the two numbers -2 and 3 both multiply to -6 and add to 1



Now replace the middle term 1x with -2x%2B3x. Remember, -2 and 3 add to 1. So this shows us that -2x%2B3x=1x.



2x%5E2%2Bhighlight%28-2x%2B3x%29-3 Replace the second term 1x with -2x%2B3x.



%282x%5E2-2x%29%2B%283x-3%29 Group the terms into two pairs.



2x%28x-1%29%2B%283x-3%29 Factor out the GCF 2x from the first group.



2x%28x-1%29%2B3%28x-1%29 Factor out 3 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.



%282x%2B3%29%28x-1%29 Combine like terms. Or factor out the common term x-1



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Answer:



So 2%2Ax%5E2%2Bx-3 factors to %282x%2B3%29%28x-1%29.



In other words, 2%2Ax%5E2%2Bx-3=%282x%2B3%29%28x-1%29.



Note: you can check the answer by expanding %282x%2B3%29%28x-1%29 to get 2%2Ax%5E2%2Bx-3 or by graphing the original expression and the answer (the two graphs should be identical).