document.write( "Question 294050: Factor: 12^2-5m-3 \n" ); document.write( "

Algebra.Com's Answer #212156 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
I will assume you forgot the m^2
\n" ); document.write( "12m^2-5m-3
\n" ); document.write( "We want factors of -36 which add up to -5
\n" ); document.write( "-9 and +4 come to mind\r
\n" ); document.write( "\n" ); document.write( "12m^2-5m-3
\n" ); document.write( "12m^2+4m-9m-3
\n" ); document.write( "4m*(3m+1)+ (-3)(3m+1)
\n" ); document.write( "(3m+1)*(4m-3)
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"12m%5E2-5m-3\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-3\"$ .

Now multiply the first coefficient $\"12\"$ by the last term $\"-3\"$ to get $\"%2812%29%28-3%29=-36\"$ .

Now the question is: what two whole numbers multiply to $\"-36\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?

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 $\"-5\"$ :

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First Number	Second Number	Sum
1	-36	1+(-36)=-35
2	-18	2+(-18)=-16
3	-12	3+(-12)=-9
4	-9	4+(-9)=-5
6	-6	6+(-6)=0
-1	36	-1+36=35
-2	18	-2+18=16
-3	12	-3+12=9
-4	9	-4+9=5
-6	6	-6+6=0

From the table, we can see that the two numbers $\"4\"$ and $\"-9\"$ add to $\"-5\"$ (the middle coefficient).

So the two numbers $\"4\"$ and $\"-9\"$ both multiply to $\"-36\"$ and add to $\"-5\"$

Now replace the middle term $\"-5m\"$ with $\"4m-9m\"$ . Remember, $\"4\"$ and $\"-9\"$ add to $\"-5\"$ . So this shows us that $\"4m-9m=-5m\"$ .

$\"12m%5E2%2Bhighlight%284m-9m%29-3\"$ Replace the second term $\"-5m\"$ with $\"4m-9m\"$ .

$\"%2812m%5E2%2B4m%29%2B%28-9m-3%29\"$ Group the terms into two pairs.

$\"4m%283m%2B1%29%2B%28-9m-3%29\"$ Factor out the GCF $\"4m\"$ from the first group.

$\"4m%283m%2B1%29-3%283m%2B1%29\"$ 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.

$\"%284m-3%29%283m%2B1%29\"$ Combine like terms. Or factor out the common term $\"3m%2B1\"$

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Answer:

So $\"12%2Am%5E2-5%2Am-3\"$ factors to $\"%284m-3%29%283m%2B1%29\"$ .

In other words, $\"12%2Am%5E2-5%2Am-3=%284m-3%29%283m%2B1%29\"$ .

Note: you can check the answer by expanding $\"%284m-3%29%283m%2B1%29\"$ to get $\"12%2Am%5E2-5%2Am-3\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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