Question 282408: Factor completely. Remember to look first for a common factor. Check by multiplying. If a polynomial is prime, state this.
27m^2 - 36 + 12
This is what I have got so far if I am heading in the right direction please let me know and what I do next if not help me figure this one out.
27m^2 - 36 + 12
=3(9x^2 - 12x + 4)
=3[(3x)^2 - 4*3 + 2^2]
That's all I got so far.
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! first i suspect that you copied the problem incorrectly
27m^2 - 36 + 12
I suspect it should be
27m^2 - 36m + 12
You perform some magic with the equation changing m's to x's
and then the last line is completely a mystery what you are doing and why you are doing it.
assuming it should be
27m^2 - 36m + 12
3*(9m^2-12m+4)
| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36
6*6 = 36
(-1)*(-36) = 36
(-2)*(-18) = 36
(-3)*(-12) = 36
(-4)*(-9) = 36
(-6)*(-6) = 36
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | 36 | 1+36=37 | | 2 | 18 | 2+18=20 | | 3 | 12 | 3+12=15 | | 4 | 9 | 4+9=13 | | 6 | 6 | 6+6=12 | | -1 | -36 | -1+(-36)=-37 | | -2 | -18 | -2+(-18)=-20 | | -3 | -12 | -3+(-12)=-15 | | -4 | -9 | -4+(-9)=-13 | | -6 | -6 | -6+(-6)=-12 |
From the table, we can see that the two numbers  and add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out  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.
 Combine like terms. Or factor out the common term 
 Condense the terms.
===============================================================
Answer:
So  factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
 Start with the given expression.
 Factor out the GCF  .
Now let's try to factor the inner expression
---------------------------------------------------------------
Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .
Now multiply the first coefficient by the last term to get .
Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36
6*6 = 36
(-1)*(-36) = 36
(-2)*(-18) = 36
(-3)*(-12) = 36
(-4)*(-9) = 36
(-6)*(-6) = 36
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | 36 | 1+36=37 | | 2 | 18 | 2+18=20 | | 3 | 12 | 3+12=15 | | 4 | 9 | 4+9=13 | | 6 | 6 | 6+6=12 | | -1 | -36 | -1+(-36)=-37 | | -2 | -18 | -2+(-18)=-20 | | -3 | -12 | -3+(-12)=-15 | | -4 | -9 | -4+(-9)=-13 | | -6 | -6 | -6+(-6)=-12 |
From the table, we can see that the two numbers and add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term with . Remember, and add to . So this shows us that .
Replace the second term with .
Group the terms into two pairs.
Factor out the GCF from the first group.
Factor out 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.
Combine like terms. Or factor out the common term
Condense the terms.
--------------------------------------------------
So then factors further to 
===============================================================
Answer:
So completely factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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