SOLUTION: reverse foil or grouping 5x^2+11x-12

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Question 299374: reverse foil or grouping
5x^2+11x-12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 5x%5E2%2B11x-12, we can see that the first coefficient is 5, the second coefficient is 11, and the last term is -12.


Now multiply the first coefficient 5 by the last term -12 to get %285%29%28-12%29=-60.


Now the question is: what two whole numbers multiply to -60 (the previous product) and add to the second coefficient 11?


To find these two numbers, we need to list all of the factors of -60 (the previous product).


Factors of -60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60


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


These factors pair up and multiply to -60.
1*(-60) = -60
2*(-30) = -60
3*(-20) = -60
4*(-15) = -60
5*(-12) = -60
6*(-10) = -60
(-1)*(60) = -60
(-2)*(30) = -60
(-3)*(20) = -60
(-4)*(15) = -60
(-5)*(12) = -60
(-6)*(10) = -60

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 11:


First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4



From the table, we can see that the two numbers -4 and 15 add to 11 (the middle coefficient).


So the two numbers -4 and 15 both multiply to -60 and add to 11


Now replace the middle term 11x with -4x%2B15x. Remember, -4 and 15 add to 11. So this shows us that -4x%2B15x=11x.


5x%5E2%2Bhighlight%28-4x%2B15x%29-12 Replace the second term 11x with -4x%2B15x.


%285x%5E2-4x%29%2B%2815x-12%29 Group the terms into two pairs.


x%285x-4%29%2B%2815x-12%29 Factor out the GCF x from the first group.


x%285x-4%29%2B3%285x-4%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.


%28x%2B3%29%285x-4%29 Combine like terms. Or factor out the common term 5x-4


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Answer:


So 5x%5E2%2B11x-12 factors to %28x%2B3%29%285x-4%29.


In other words, 5x%5E2%2B11x-12=%28x%2B3%29%285x-4%29.


Note: you can check the answer by expanding %28x%2B3%29%285x-4%29 to get 5x%5E2%2B11x-12 or by graphing the original expression and the answer (the two graphs should be identical).