document.write( "Question 816885: how to factor\r
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Algebra.Com's Answer #491764 by richwmiller(17219) $\"\"$ $\"About$

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

Looking at the expression $\"3x%5E2-13x%2B4\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"4\"$ .

Now multiply the first coefficient $\"3\"$ by the last term $\"4\"$ to get $\"%283%29%284%29=12\"$ .

Now the question is: what two whole numbers multiply to $\"12\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?

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 $\"-13\"$ :

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First Number	Second Number	Sum
1	12	1+12=13
2	6	2+6=8
3	4	3+4=7
-1	-12	-1+(-12)=-13
-2	-6	-2+(-6)=-8
-3	-4	-3+(-4)=-7

From the table, we can see that the two numbers $\"-1\"$ and $\"-12\"$ add to $\"-13\"$ (the middle coefficient).

So the two numbers $\"-1\"$ and $\"-12\"$ both multiply to $\"12\"$ and add to $\"-13\"$

Now replace the middle term $\"-13x\"$ with $\"-x-12x\"$ . Remember, $\"-1\"$ and $\"-12\"$ add to $\"-13\"$ . So this shows us that $\"-x-12x=-13x\"$ .

$\"3x%5E2%2Bhighlight%28-x-12x%29%2B4\"$ Replace the second term $\"-13x\"$ with $\"-x-12x\"$ .

$\"%283x%5E2-x%29%2B%28-12x%2B4%29\"$ Group the terms into two pairs.

$\"x%283x-1%29%2B%28-12x%2B4%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%283x-1%29-4%283x-1%29\"$ Factor out $\"4\"$ 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.

$\"%28x-4%29%283x-1%29\"$ Combine like terms. Or factor out the common term $\"3x-1\"$

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Answer:

So $\"3%2Ax%5E2-13%2Ax%2B4\"$ factors to $\"%28x-4%29%283x-1%29\"$ .

In other words, $\"3%2Ax%5E2-13%2Ax%2B4=%28x-4%29%283x-1%29\"$ .

Note: you can check the answer by expanding $\"%28x-4%29%283x-1%29\"$ to get $\"3%2Ax%5E2-13%2Ax%2B4\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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