SOLUTION: Factor completely.
252r^2 + 588rf + 343f^2
Thank you for any help you can give me in figuring this problem out.
Algebra.Com
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) (Show Source): You can put this solution on YOUR website!
Start with the given expression
Factor out the GCF
Now let's focus on the inner expression
------------------------------------------------------------
Looking at we can see that the first term is and the last term is 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 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 this list we can see that 42 and 42 add up to 84 and multiply to 1764
Now looking at the expression , replace with (notice adds up to . So it is equivalent to )
Now let's factor by grouping:
Group like terms
Factor out the GCF of out of the first group. Factor out the GCF of out of the second group
Since we have a common term of , we can combine like terms
So factors to
So this also means that factors to (since is equivalent to )
note: is equivalent to since the term occurs twice. So also factors to
------------------------------------------------------------
So our expression goes from and factors further to
------------------
Answer:
So factors to