SOLUTION: Factor completely. 252r^2 + 588rf + 343f^2 Thank you for any help you can give me in figuring this problem out.

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Question 133108: Factor completely.
252r^2 + 588rf + 343f^2
Thank you for any help you can give me in figuring this problem out.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

252r%5E2%2B588rf%2B343f%5E2 Start with the given expression


7%2836r%5E2%2B84rf%2B49f%5E2%29 Factor out the GCF 7


Now let's focus on the inner expression 36r%5E2%2B84rf%2B49f%5E2




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Looking at 36r%5E2%2B84rf%2B49f%5E2 we can see that the first term is 36r%5E2 and the last term is 49f%5E2 where the coefficients are 36 and 49 respectively.

Now multiply the first coefficient 36 and the last coefficient 49 to get 1764. Now what two numbers multiply to 1764 and add to the middle coefficient 84? Let's list all of the factors of 1764:



Factors of 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

-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 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 1764
1*1764
2*882
3*588
4*441
6*294
7*252
9*196
12*147
14*126
18*98
21*84
28*63
36*49
42*42
(-1)*(-1764)
(-2)*(-882)
(-3)*(-588)
(-4)*(-441)
(-6)*(-294)
(-7)*(-252)
(-9)*(-196)
(-12)*(-147)
(-14)*(-126)
(-18)*(-98)
(-21)*(-84)
(-28)*(-63)
(-36)*(-49)
(-42)*(-42)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 84? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 84

First NumberSecond NumberSum
117641+1764=1765
28822+882=884
35883+588=591
44414+441=445
62946+294=300
72527+252=259
91969+196=205
1214712+147=159
1412614+126=140
189818+98=116
218421+84=105
286328+63=91
364936+49=85
424242+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 this list we can see that 42 and 42 add up to 84 and multiply to 1764


Now looking at the expression 36r%5E2%2B84rf%2B49f%5E2, replace 84rf with 42rf%2B42rf (notice 42rf%2B42rf adds up to 84rf. So it is equivalent to 84rf)

36r%5E2%2Bhighlight%2842rf%2B42rf%29%2B49f%5E2


Now let's factor 36r%5E2%2B42rf%2B42rf%2B49f%5E2 by grouping:


%2836r%5E2%2B42rf%29%2B%2842rf%2B49f%5E2%29 Group like terms


6r%286r%2B7f%29%2B7f%286r%2B7f%29 Factor out the GCF of 6r out of the first group. Factor out the GCF of 7f out of the second group


%286r%2B7f%29%286r%2B7f%29 Since we have a common term of 6r%2B7f, we can combine like terms

So 36r%5E2%2B42rf%2B42rf%2B49f%5E2 factors to %286r%2B7f%29%286r%2B7f%29


So this also means that 36r%5E2%2B84rf%2B49f%5E2 factors to %286r%2B7f%29%286r%2B7f%29 (since 36r%5E2%2B84rf%2B49f%5E2 is equivalent to 36r%5E2%2B42rf%2B42rf%2B49f%5E2)


note: %286r%2B7f%29%286r%2B7f%29 is equivalent to %286r%2B7f%29%5E2 since the term 6r%2B7f occurs twice. So 36r%5E2%2B84rf%2B49f%5E2 also factors to %286r%2B7f%29%5E2



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So our expression goes from 7%2836r%5E2%2B84rf%2B49f%5E2%29 and factors further to 7%286r%2B7f%29%5E2


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Answer:

So 252r%5E2%2B588rf%2B343f%5E2 factors to 7%286r%2B7f%29%5E2