SOLUTION: Can someone tell me how to factor polynomials by grouping, such as: {{{36a^2-36ab-27b^2}}} This is part of my take home final and is really stressing me out!

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Question 387326: Can someone tell me how to factor polynomials by grouping, such as:
36a%5E2-36ab-27b%5E2
This is part of my take home final and is really stressing me out!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

36a%5E2-36ab-27b%5E2 Start with the given expression.


9%284a%5E2-4ab-3b%5E2%29 Factor out the GCF 9.


Now let's try to factor the inner expression 4a%5E2-4ab-3b%5E2


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Looking at the expression 4a%5E2-4ab-3b%5E2, we can see that the first coefficient is 4, the second coefficient is -4, and the last coefficient is -3.


Now multiply the first coefficient 4 by the last coefficient -3 to get %284%29%28-3%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 -4ab with 2ab-6ab. Remember, 2 and -6 add to -4. So this shows us that 2ab-6ab=-4ab.


4a%5E2%2Bhighlight%282ab-6ab%29-3b%5E2 Replace the second term -4ab with 2ab-6ab.


%284a%5E2%2B2ab%29%2B%28-6ab-3b%5E2%29 Group the terms into two pairs.


2a%282a%2Bb%29%2B%28-6ab-3b%5E2%29 Factor out the GCF 2a from the first group.


2a%282a%2Bb%29-3b%282a%2Bb%29 Factor out -3b from the second group.


%282a-3b%29%282a%2Bb%29 Factor 2a%2Bb from the entire expression.


So 4a%5E2-4ab-3b%5E2 factors to %282a-3b%29%282a%2Bb%29


This means that 36a%5E2-36ab-27b%5E2 completely factors to 9%282a-3b%29%282a%2Bb%29


In other words, 36a%5E2-36ab-27b%5E2=9%282a-3b%29%282a%2Bb%29


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim