document.write( "Question 407821: I am needing help please. \r
\n" ); document.write( "\n" ); document.write( "Completly factor the expression and show all work: \r
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Algebra.Com's Answer #287386 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"245x%5E3-420x%5E2y%2B180xy%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%2849x%5E2-84xy%2B36y%5E2%29\"$ Factor out the GCF $\"5x\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"49x%5E2-84xy%2B36y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"49x%5E2-84xy%2B36y%5E2\"$ we can see that the first term is $\"49x%5E2\"$ and the last term is $\"36y%5E2\"$ where the coefficients are 49 and 36 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 49 and the last coefficient 36 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 $\"49x%5E2-84xy%2B36y%5E2\"$ , replace $\"-84xy\"$ with $\"-42xy%2B-42xy\"$ (notice $\"-42xy%2B-42xy\"$ adds up to $\"-84xy\"$ . So it is equivalent to $\"-84xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"49x%5E2%2Bhighlight%28-42xy%2B-42xy%29%2B36y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"49x%5E2-42xy-42xy%2B36y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2849x%5E2-42xy%29%2B%28-42xy%2B36y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"7x%287x-6y%29-6y%287x-6y%29\"$ Factor out the GCF of $\"7x\"$ out of the first group. Factor out the GCF of $\"-6y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%287x-6y%29%287x-6y%29\"$ Since we have a common term of $\"7x-6y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"49x%5E2-42xy-42xy%2B36y%5E2\"$ factors to $\"%287x-6y%29%287x-6y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"49x%5E2-84xy%2B36y%5E2\"$ factors to $\"%287x-6y%29%287x-6y%29\"$ (since $\"49x%5E2-84xy%2B36y%5E2\"$ is equivalent to $\"49x%5E2-42xy-42xy%2B36y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%287x-6y%29%287x-6y%29\"$ is equivalent to $\"%287x-6y%29%5E2\"$ since the term $\"7x-6y\"$ occurs twice. So $\"49x%5E2-84xy%2B36y%5E2\"$ also factors to $\"%287x-6y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"5x%2849x%5E2-84xy%2B36y%5E2%29\"$ and factors further to $\"5x%287x-6y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"245x%5E3-420x%5E2y%2B180xy%5E2\"$ factors to $\"5x%287x-6y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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