SOLUTION: having problem with factoring completely, the problems are 3x^3-12x and 128x^2-224x+98

Question 114573: having problem with factoring completely, the problems are 3x^3-12x and 128x^2-224x+98
Found 2 solutions by jim_thompson5910, checkley71:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
#1
Start with the given expression

Factor out the GCF 3x

Factor using the difference of squares

#2

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

Start with the given expression.

Factor out the GCF .

Now let's try to factor the inner expression

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Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

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

Factors of :

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 .

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 :

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 and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out 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.

Combine like terms. Or factor out the common term

Condense the terms.

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So then factors further to

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Answer:

So completely factors to .

In other words, .

Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

Answer by checkley71(8403) (Show Source): You can put this solution on YOUR website!
3X^3-12X
3X(X^2-4)
3X(X+2)(X-2)
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128X^2-224X+98
2(64X^2-112X+49)
2(8X-7)(8X-7)