document.write( "Question 339939: Factor Completely\r
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\n" ); document.write( "\n" ); document.write( "c^3 + 20c^2 + 100c \n" ); document.write( "

Algebra.Com's Answer #243604 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"c%5E3%2B20c%5E2%2B100c\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"c%28c%5E2%2B20c%2B100%29\"$ Factor out the GCF $\"c\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"c%5E2%2B20c%2B100\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"1c%5E2%2B20c%2B100\"$ we can see that the first term is $\"1c%5E2\"$ and the last term is $\"100\"$ where the coefficients are 1 and 100 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 100 to get 100. Now what two numbers multiply to 100 and add to the middle coefficient 20? Let's list all of the factors of 100:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 100:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20,-25,-50 ...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 100\r
\n" ); document.write( "\n" ); document.write( "1*100\r
\n" ); document.write( "\n" ); document.write( "2*50\r
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\n" ); document.write( "\n" ); document.write( "10*10\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-100)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-50)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-25)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-20)\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 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20\r
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First Number	Second Number	Sum
1	100	1+100=101
2	50	2+50=52
4	25	4+25=29
5	20	5+20=25
10	10	10+10=20
-1	-100	-1+(-100)=-101
-2	-50	-2+(-50)=-52
-4	-25	-4+(-25)=-29
-5	-20	-5+(-20)=-25
-10	-10	-10+(-10)=-20

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 10 and 10 add up to 20 and multiply to 100\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"1c%5E2%2B20c%2B100\"$ , replace $\"20c\"$ with $\"10c%2B10c\"$ (notice $\"10c%2B10c\"$ adds up to $\"20c\"$ . So it is equivalent to $\"20c\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"1c%5E2%2Bhighlight%2810c%2B10c%29%2B100\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1c%5E2%2B10c%2B10c%2B100\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%281c%5E2%2B10c%29%2B%2810c%2B100%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"c%28c%2B10%29%2B10%28c%2B10%29\"$ Factor out the GCF of $\"c\"$ out of the first group. Factor out the GCF of $\"10\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28c%2B10%29%28c%2B10%29\"$ Since we have a common term of $\"c%2B10\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"1c%5E2%2B10c%2B10c%2B100\"$ factors to $\"%28c%2B10%29%28c%2B10%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"1c%5E2%2B20c%2B100\"$ factors to $\"%28c%2B10%29%28c%2B10%29\"$ (since $\"1c%5E2%2B20c%2B100\"$ is equivalent to $\"1c%5E2%2B10c%2B10c%2B100\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%28c%2B10%29%28c%2B10%29\"$ is equivalent to $\"%28c%2B10%29%5E2\"$ since the term $\"c%2B10\"$ occurs twice. So $\"1c%5E2%2B20c%2B100\"$ also factors to $\"%28c%2B10%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"c%28c%5E2%2B20c%2B100%29\"$ and factors further to $\"c%28c%2B10%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"c%5E3%2B20c%5E2%2B100c\"$ factors to $\"c%28c%2B10%29%5E2\"$ \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, feel free to check out my tutoring website\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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