document.write( "Question 282408: Factor completely. Remember to look first for a common factor. Check by multiplying. If a polynomial is prime, state this.\r
\n" ); document.write( "\n" ); document.write( "27m^2 - 36 + 12\r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "27m^2 - 36 + 12
\n" ); document.write( "=3(9x^2 - 12x + 4)
\n" ); document.write( "=3[(3x)^2 - 4*3 + 2^2]\r
\n" ); document.write( "\n" ); document.write( "That's all I got so far. \n" ); document.write( "

Algebra.Com's Answer #205046 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
first i suspect that you copied the problem incorrectly\r
\n" ); document.write( "\n" ); document.write( "27m^2 - 36 + 12
\n" ); document.write( "I suspect it should be
\n" ); document.write( "27m^2 - 36m + 12 \r
\n" ); document.write( "\n" ); document.write( "You perform some magic with the equation changing m's to x's
\n" ); document.write( "and then the last line is completely a mystery what you are doing and why you are doing it.
\n" ); document.write( "assuming it should be
\n" ); document.write( "27m^2 - 36m + 12
\n" ); document.write( "3*(9m^2-12m+4)\r
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"9m%5E2-12m%2B4\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"-12\"$ , and the last term is $\"4\"$ .

Now multiply the first coefficient $\"9\"$ by the last term $\"4\"$ to get $\"%289%29%284%29=36\"$ .

Now the question is: what two whole numbers multiply to $\"36\"$ (the previous product) and add to the second coefficient $\"-12\"$ ?

To find these two numbers, we need to list all of the factors of $\"36\"$ (the previous product).

Factors of $\"36\"$ :

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 $\"36\"$ .

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 $\"-12\"$ :

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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 $\"-6\"$ and $\"-6\"$ add to $\"-12\"$ (the middle coefficient).

So the two numbers $\"-6\"$ and $\"-6\"$ both multiply to $\"36\"$ and add to $\"-12\"$

Now replace the middle term $\"-12m\"$ with $\"-6m-6m\"$ . Remember, $\"-6\"$ and $\"-6\"$ add to $\"-12\"$ . So this shows us that $\"-6m-6m=-12m\"$ .

$\"9m%5E2%2Bhighlight%28-6m-6m%29%2B4\"$ Replace the second term $\"-12m\"$ with $\"-6m-6m\"$ .

$\"%289m%5E2-6m%29%2B%28-6m%2B4%29\"$ Group the terms into two pairs.

$\"3m%283m-2%29%2B%28-6m%2B4%29\"$ Factor out the GCF $\"3m\"$ from the first group.

$\"3m%283m-2%29-2%283m-2%29\"$ Factor out $\"2\"$ 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.

$\"%283m-2%29%283m-2%29\"$ Combine like terms. Or factor out the common term $\"3m-2\"$

$\"%283m-2%29%5E2\"$ Condense the terms.

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

Answer:

So $\"9%2Am%5E2-12%2Am%2B4\"$ factors to $\"%283m-2%29%5E2\"$ .

In other words, $\"9%2Am%5E2-12%2Am%2B4=%283m-2%29%5E2\"$ .

Note: you can check the answer by expanding $\"%283m-2%29%5E2\"$ to get $\"9%2Am%5E2-12%2Am%2B4\"$ 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)

$\"27%2Am%5E2-36%2Am%2B12\"$ Start with the given expression.

$\"3%289m%5E2-12m%2B4%29\"$ Factor out the GCF $\"3\"$ .

Now let's try to factor the inner expression $\"9m%5E2-12m%2B4\"$

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

Looking at the expression $\"9m%5E2-12m%2B4\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"-12\"$ , and the last term is $\"4\"$ .

Now multiply the first coefficient $\"9\"$ by the last term $\"4\"$ to get $\"%289%29%284%29=36\"$ .

Now the question is: what two whole numbers multiply to $\"36\"$ (the previous product) and add to the second coefficient $\"-12\"$ ?

To find these two numbers, we need to list all of the factors of $\"36\"$ (the previous product).

Factors of $\"36\"$ :

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 $\"36\"$ .

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 $\"-12\"$ :

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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 $\"-6\"$ and $\"-6\"$ add to $\"-12\"$ (the middle coefficient).

So the two numbers $\"-6\"$ and $\"-6\"$ both multiply to $\"36\"$ and add to $\"-12\"$

Now replace the middle term $\"-12m\"$ with $\"-6m-6m\"$ . Remember, $\"-6\"$ and $\"-6\"$ add to $\"-12\"$ . So this shows us that $\"-6m-6m=-12m\"$ .

$\"9m%5E2%2Bhighlight%28-6m-6m%29%2B4\"$ Replace the second term $\"-12m\"$ with $\"-6m-6m\"$ .

$\"%289m%5E2-6m%29%2B%28-6m%2B4%29\"$ Group the terms into two pairs.

$\"3m%283m-2%29%2B%28-6m%2B4%29\"$ Factor out the GCF $\"3m\"$ from the first group.

$\"3m%283m-2%29-2%283m-2%29\"$ Factor out $\"2\"$ 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.

$\"%283m-2%29%283m-2%29\"$ Combine like terms. Or factor out the common term $\"3m-2\"$

$\"%283m-2%29%5E2\"$ Condense the terms.

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

So $\"3%289m%5E2-12m%2B4%29\"$ then factors further to $\"3%283m-2%29%5E2\"$

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

Answer:

So $\"27%2Am%5E2-36%2Am%2B12\"$ completely factors to $\"3%283m-2%29%5E2\"$ .

In other words, $\"27%2Am%5E2-36%2Am%2B12=3%283m-2%29%5E2\"$ .

Note: you can check the answer by expanding $\"3%283m-2%29%5E2\"$ to get $\"27%2Am%5E2-36%2Am%2B12\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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