document.write( "Question 133108: Factor completely.\r
\n" ); document.write( "\n" ); document.write( "252r^2 + 588rf + 343f^2\r
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Algebra.Com's Answer #97318 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"252r%5E2%2B588rf%2B343f%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"7%2836r%5E2%2B84rf%2B49f%5E2%29\"$ Factor out the GCF $\"7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"36r%5E2%2B84rf%2B49f%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "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:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 1764:\r
\n" ); document.write( "\n" ); document.write( "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\r
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\n" ); document.write( "\n" ); document.write( "-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\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to 1764\r
\n" ); document.write( "\n" ); document.write( "1*1764\r
\n" ); document.write( "\n" ); document.write( "2*882\r
\n" ); document.write( "\n" ); document.write( "3*588\r
\n" ); document.write( "\n" ); document.write( "4*441\r
\n" ); document.write( "\n" ); document.write( "6*294\r
\n" ); document.write( "\n" ); document.write( "7*252\r
\n" ); document.write( "\n" ); document.write( "9*196\r
\n" ); document.write( "\n" ); document.write( "12*147\r
\n" ); document.write( "\n" ); document.write( "14*126\r
\n" ); document.write( "\n" ); document.write( "18*98\r
\n" ); document.write( "\n" ); document.write( "21*84\r
\n" ); document.write( "\n" ); document.write( "28*63\r
\n" ); document.write( "\n" ); document.write( "36*49\r
\n" ); document.write( "\n" ); document.write( "42*42\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-1764)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-882)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-588)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-441)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-294)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(-252)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-196)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(-147)\r
\n" ); document.write( "\n" ); document.write( "(-14)*(-126)\r
\n" ); document.write( "\n" ); document.write( "(-18)*(-98)\r
\n" ); document.write( "\n" ); document.write( "(-21)*(-84)\r
\n" ); document.write( "\n" ); document.write( "(-28)*(-63)\r
\n" ); document.write( "\n" ); document.write( "(-36)*(-49)\r
\n" ); document.write( "\n" ); document.write( "(-42)*(-42)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "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\r
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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

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 42 and 42 add up to 84 and multiply to 1764\r
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\n" ); document.write( "\n" ); document.write( "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 $\"84rf\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"36r%5E2%2Bhighlight%2842rf%2B42rf%29%2B49f%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"36r%5E2%2B42rf%2B42rf%2B49f%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2836r%5E2%2B42rf%29%2B%2842rf%2B49f%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"6r%286r%2B7f%29%2B7f%286r%2B7f%29\"$ Factor out the GCF of $\"6r\"$ out of the first group. Factor out the GCF of $\"7f\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%286r%2B7f%29%286r%2B7f%29\"$ Since we have a common term of $\"6r%2B7f\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"36r%5E2%2B42rf%2B42rf%2B49f%5E2\"$ factors to $\"%286r%2B7f%29%286r%2B7f%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "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 $\"36r%5E2%2B42rf%2B42rf%2B49f%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "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 $\"%286r%2B7f%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"7%2836r%5E2%2B84rf%2B49f%5E2%29\"$ and factors further to $\"7%286r%2B7f%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"252r%5E2%2B588rf%2B343f%5E2\"$ factors to $\"7%286r%2B7f%29%5E2\"$ \r
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