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Question 506441: Please Help Me solve This problem: factor 252w^2+588wf+343f^2
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
 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 coefficient is  .
Now multiply the first coefficient by the last coefficient 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 :
1,2,3,4,6,7,9,12,14,18,21,28,36,42,49,63,84,98,126,147,196,252,294,441,588,882,1764
-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-49,-63,-84,-98,-126,-147,-196,-252,-294,-441,-588,-882,-1764
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*1764 = 1764
2*882 = 1764
3*588 = 1764
4*441 = 1764
6*294 = 1764
7*252 = 1764
9*196 = 1764
12*147 = 1764
14*126 = 1764
18*98 = 1764
21*84 = 1764
28*63 = 1764
36*49 = 1764
42*42 = 1764
(-1)*(-1764) = 1764
(-2)*(-882) = 1764
(-3)*(-588) = 1764
(-4)*(-441) = 1764
(-6)*(-294) = 1764
(-7)*(-252) = 1764
(-9)*(-196) = 1764
(-12)*(-147) = 1764
(-14)*(-126) = 1764
(-18)*(-98) = 1764
(-21)*(-84) = 1764
(-28)*(-63) = 1764
(-36)*(-49) = 1764
(-42)*(-42) = 1764
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 | 1764 | 1+1764=1765 | | 2 | 882 | 2+882=884 | | 3 | 588 | 3+588=591 | | 4 | 441 | 4+441=445 | | 6 | 294 | 6+294=300 | | 7 | 252 | 7+252=259 | | 9 | 196 | 9+196=205 | | 12 | 147 | 12+147=159 | | 14 | 126 | 14+126=140 | | 18 | 98 | 18+98=116 | | 21 | 84 | 21+84=105 | | 28 | 63 | 28+63=91 | | 36 | 49 | 36+49=85 | | 42 | 42 | 42+42=84 | | -1 | -1764 | -1+(-1764)=-1765 | | -2 | -882 | -2+(-882)=-884 | | -3 | -588 | -3+(-588)=-591 | | -4 | -441 | -4+(-441)=-445 | | -6 | -294 | -6+(-294)=-300 | | -7 | -252 | -7+(-252)=-259 | | -9 | -196 | -9+(-196)=-205 | | -12 | -147 | -12+(-147)=-159 | | -14 | -126 | -14+(-126)=-140 | | -18 | -98 | -18+(-98)=-116 | | -21 | -84 | -21+(-84)=-105 | | -28 | -63 | -28+(-63)=-91 | | -36 | -49 | -36+(-49)=-85 | | -42 | -42 | -42+(-42)=-84 |
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).
Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at jim_thompson5910@hotmail.com
or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html
Thanks,
Jim
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