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

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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) About Me  (Show Source):
You can put this solution on YOUR website!
#1
3x%5E3-12x Start with the given expression


3x%28x%5E2-4%29 Factor out the GCF 3x


3x%28x-2%29%28x%2B2%29 Factor x%5E2-4 using the difference of squares



#2
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:



First NumberSecond NumberSum
131361+3136=3137
215682+1568=1570
47844+784=788
74487+448=455
83928+392=400
1422414+224=238
1619616+196=212
2811228+112=140
329832+98=130
496449+64=113
565656+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).

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