SOLUTION: reverse foil or grouping 3x^2+12x-36

Algebra ->  Divisibility and Prime Numbers -> SOLUTION: reverse foil or grouping 3x^2+12x-36      Log On


   

Question 299372: reverse foil or grouping
3x^2+12x-36
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

3x%5E2%2B12x-36 Start with the given expression.


3%28x%5E2%2B4x-12%29 Factor out the GCF 3.


Now let's try to factor the inner expression x%5E2%2B4x-12


---------------------------------------------------------------


Looking at the expression x%5E2%2B4x-12, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is -12.


Now multiply the first coefficient 1 by the last term -12 to get %281%29%28-12%29=-12.


Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient 4?


To find these two numbers, we need to list all of the factors of -12 (the previous product).


Factors of -12:
1,2,3,4,6,12
-1,-2,-3,-4,-6,-12


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


These factors pair up and multiply to -12.
1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:


First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1



From the table, we can see that the two numbers -2 and 6 add to 4 (the middle coefficient).


So the two numbers -2 and 6 both multiply to -12 and add to 4


Now replace the middle term 4x with -2x%2B6x. Remember, -2 and 6 add to 4. So this shows us that -2x%2B6x=4x.


x%5E2%2Bhighlight%28-2x%2B6x%29-12 Replace the second term 4x with -2x%2B6x.


%28x%5E2-2x%29%2B%286x-12%29 Group the terms into two pairs.


x%28x-2%29%2B%286x-12%29 Factor out the GCF x from the first group.


x%28x-2%29%2B6%28x-2%29 Factor out 6 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%2B6%29%28x-2%29 Combine like terms. Or factor out the common term x-2


--------------------------------------------------


So 3%28x%5E2%2B4x-12%29 then factors further to 3%28x%2B6%29%28x-2%29


===============================================================


Answer:


So 3x%5E2%2B12x-36 completely factors to 3%28x%2B6%29%28x-2%29.


In other words, 3x%5E2%2B12x-36=3%28x%2B6%29%28x-2%29.


Note: you can check the answer by expanding 3%28x%2B6%29%28x-2%29 to get 3x%5E2%2B12x-36 or by graphing the original expression and the answer (the two graphs should be identical).