document.write( "Question 114573: having problem with factoring completely, the problems are 3x^3-12x and 128x^2-224x+98 \n" ); document.write( "

Algebra.Com's Answer #83346 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"3x%5E3-12x\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%28x-2%29%28x%2B2%29\"$ Factor $\"x%5E2-4\"$ using the difference of squares\r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"128%2Ax%5E2-224%2Ax%2B98\"$ Start with the given expression.

$\"2%2864x%5E2-112x%2B49%29\"$ Factor out the GCF $\"2\"$ .

Now let's try to factor the inner expression $\"64x%5E2-112x%2B49\"$

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Looking at the expression $\"64x%5E2-112x%2B49\"$ , we can see that the first coefficient is $\"64\"$ , the second coefficient is $\"-112\"$ , and the last term is $\"49\"$ .

Now multiply the first coefficient $\"64\"$ by the last term $\"49\"$ to get $\"%2864%29%2849%29=3136\"$ .

Now the question is: what two whole numbers multiply to $\"3136\"$ (the previous product) and add to the second coefficient $\"-112\"$ ?

To find these two numbers, we need to list all of the factors of $\"3136\"$ (the previous product).

Factors of $\"3136\"$ :

1,2,4,7,8,14,16,28,32,49,56,64,98,112,196,224,392,448,784,1568,3136

-1,-2,-4,-7,-8,-14,-16,-28,-32,-49,-56,-64,-98,-112,-196,-224,-392,-448,-784,-1568,-3136

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

These factors pair up and multiply to $\"3136\"$ .

1*3136 = 3136
2*1568 = 3136
4*784 = 3136
7*448 = 3136
8*392 = 3136
14*224 = 3136
16*196 = 3136
28*112 = 3136
32*98 = 3136
49*64 = 3136
56*56 = 3136
(-1)*(-3136) = 3136
(-2)*(-1568) = 3136
(-4)*(-784) = 3136
(-7)*(-448) = 3136
(-8)*(-392) = 3136
(-14)*(-224) = 3136
(-16)*(-196) = 3136
(-28)*(-112) = 3136
(-32)*(-98) = 3136
(-49)*(-64) = 3136
(-56)*(-56) = 3136

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-112\"$ :

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First Number	Second Number	Sum
1	3136	1+3136=3137
2	1568	2+1568=1570
4	784	4+784=788
7	448	7+448=455
8	392	8+392=400
14	224	14+224=238
16	196	16+196=212
28	112	28+112=140
32	98	32+98=130
49	64	49+64=113
56	56	56+56=112
-1	-3136	-1+(-3136)=-3137
-2	-1568	-2+(-1568)=-1570
-4	-784	-4+(-784)=-788
-7	-448	-7+(-448)=-455
-8	-392	-8+(-392)=-400
-14	-224	-14+(-224)=-238
-16	-196	-16+(-196)=-212
-28	-112	-28+(-112)=-140
-32	-98	-32+(-98)=-130
-49	-64	-49+(-64)=-113
-56	-56	-56+(-56)=-112

From the table, we can see that the two numbers $\"-56\"$ and $\"-56\"$ add to $\"-112\"$ (the middle coefficient).

So the two numbers $\"-56\"$ and $\"-56\"$ both multiply to $\"3136\"$ and add to $\"-112\"$

Now replace the middle term $\"-112x\"$ with $\"-56x-56x\"$ . Remember, $\"-56\"$ and $\"-56\"$ add to $\"-112\"$ . So this shows us that $\"-56x-56x=-112x\"$ .

$\"64x%5E2%2Bhighlight%28-56x-56x%29%2B49\"$ Replace the second term $\"-112x\"$ with $\"-56x-56x\"$ .

$\"%2864x%5E2-56x%29%2B%28-56x%2B49%29\"$ Group the terms into two pairs.

$\"8x%288x-7%29%2B%28-56x%2B49%29\"$ Factor out the GCF $\"8x\"$ from the first group.

$\"8x%288x-7%29-7%288x-7%29\"$ Factor out $\"7\"$ 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.

$\"%288x-7%29%288x-7%29\"$ Combine like terms. Or factor out the common term $\"8x-7\"$

$\"%288x-7%29%5E2\"$ Condense the terms.

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So $\"2%2864x%5E2-112x%2B49%29\"$ then factors further to $\"2%288x-7%29%5E2\"$

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Answer:

So $\"128%2Ax%5E2-224%2Ax%2B98\"$ completely factors to $\"2%288x-7%29%5E2\"$ .

In other words, $\"128%2Ax%5E2-224%2Ax%2B98=2%288x-7%29%5E2\"$ .

Note: you can check the answer by expanding $\"2%288x-7%29%5E2\"$ to get $\"128%2Ax%5E2-224%2Ax%2B98\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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