document.write( "Question 98538: Factor completely\r
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Algebra.Com's Answer #71691 by jim_thompson5910(35256) $\"\"$ $\"About$

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

Looking at the expression $\"16x%5E2-2x-3\"$ , we can see that the first coefficient is $\"16\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"-3\"$ .

Now multiply the first coefficient $\"16\"$ by the last term $\"-3\"$ to get $\"%2816%29%28-3%29=-48\"$ .

Now the question is: what two whole numbers multiply to $\"-48\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?

To find these two numbers, we need to list all of the factors of $\"-48\"$ (the previous product).

Factors of $\"-48\"$ :

1,2,3,4,6,8,12,16,24,48

-1,-2,-3,-4,-6,-8,-12,-16,-24,-48

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-48\"$ .

1*(-48) = -48
2*(-24) = -48
3*(-16) = -48
4*(-12) = -48
6*(-8) = -48
(-1)*(48) = -48
(-2)*(24) = -48
(-3)*(16) = -48
(-4)*(12) = -48
(-6)*(8) = -48

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-2\"$ :

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First Number	Second Number	Sum
1	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

From the table, we can see that the two numbers $\"6\"$ and $\"-8\"$ add to $\"-2\"$ (the middle coefficient).

So the two numbers $\"6\"$ and $\"-8\"$ both multiply to $\"-48\"$ and add to $\"-2\"$

Now replace the middle term $\"-2x\"$ with $\"6x-8x\"$ . Remember, $\"6\"$ and $\"-8\"$ add to $\"-2\"$ . So this shows us that $\"6x-8x=-2x\"$ .

$\"16x%5E2%2Bhighlight%286x-8x%29-3\"$ Replace the second term $\"-2x\"$ with $\"6x-8x\"$ .

$\"%2816x%5E2%2B6x%29%2B%28-8x-3%29\"$ Group the terms into two pairs.

$\"2x%288x%2B3%29%2B%28-8x-3%29\"$ Factor out the GCF $\"2x\"$ from the first group.

$\"2x%288x%2B3%29-1%288x%2B3%29\"$ Factor out $\"1\"$ 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.

$\"%282x-1%29%288x%2B3%29\"$ Combine like terms. Or factor out the common term $\"8x%2B3\"$

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Answer:

So $\"16%2Ax%5E2-2%2Ax-3\"$ factors to $\"%282x-1%29%288x%2B3%29\"$ .

In other words, $\"16%2Ax%5E2-2%2Ax-3=%282x-1%29%288x%2B3%29\"$ .

Note: you can check the answer by expanding $\"%282x-1%29%288x%2B3%29\"$ to get $\"16%2Ax%5E2-2%2Ax-3\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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