document.write( "Question 405542: Ok, I'm working on factoring completely. \r
\n" ); document.write( "\n" ); document.write( "Here is my equation to factor:\r
\n" ); document.write( "\n" ); document.write( "-4n^4 + 40n^3 - 100n^2 \r
\n" ); document.write( "\n" ); document.write( "that equals =\r
\n" ); document.write( "\n" ); document.write( "-4n^2 (n^2 - 10n + 25)\r
\n" ); document.write( "\n" ); document.write( "and I'm supposed to factor it down more so that the answer is
\n" ); document.write( "-4n^2(n - 5)^2\r
\n" ); document.write( "\n" ); document.write( "I get it all until the last step. How and Why am I supposed to do that?\r
\n" ); document.write( "\n" ); document.write( "Mucho thanks for any help! \n" ); document.write( "

Algebra.Com's Answer #286438 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( " $\"-4n%5E4%2B40n%5E3-100n%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"-4n%5E2%28n%5E2-10n%2B25%29\"$ Factor out the GCF $\"-4n%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"n%5E2-10n%2B25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"n%5E2-10n%2B25\"$ we can see that the first term is $\"n%5E2\"$ and the last term is $\"25\"$ where the coefficients are 1 and 25 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 25 to get 25. Now what two numbers multiply to 25 and add to the middle coefficient -10? Let's list all of the factors of 25:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 25:\r
\n" ); document.write( "\n" ); document.write( "1,5\r
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\n" ); document.write( "\n" ); document.write( "-1,-5 ...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 25\r
\n" ); document.write( "\n" ); document.write( "1*25\r
\n" ); document.write( "\n" ); document.write( "5*5\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-25)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-5)\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 -10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -10\r
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First Number	Second Number	Sum
1	25	1+25=26
5	5	5+5=10
-1	-25	-1+(-25)=-26
-5	-5	-5+(-5)=-10

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -5 and -5 add up to -10 and multiply to 25\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"n%5E2-10n%2B25\"$ , replace $\"-10n\"$ with $\"-5n-5n\"$ (notice $\"-5n-5n\"$ combines back to $\"-10n\"$ . So it is equivalent to $\"-10n\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"n%5E2%2Bhighlight%28-5n-5n%29%2B25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"n%5E2-5n-5n%2B25\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28n%5E2-5n%29%2B%28-5n%2B25%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"n%28n-5%29-5%28n-5%29\"$ Factor out the GCF of $\"n\"$ out of the first group. Factor out the GCF of $\"-5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28n-5%29%28n-5%29\"$ Since we have a common term of $\"n-5\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"n%5E2-5n-5n%2B25\"$ factors to $\"%28n-5%29%28n-5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"n%5E2-10n%2B25\"$ factors to $\"%28n-5%29%28n-5%29\"$ (since $\"n%5E2-10n%2B25\"$ is equivalent to $\"n%5E2-5n-5n%2B25\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%28n-5%29%28n-5%29\"$ is equivalent to $\"%28n-5%29%5E2\"$ since the term $\"n-5\"$ occurs twice. So $\"n%5E2-10n%2B25\"$ also factors to $\"%28n-5%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"-4n%5E2%28n%5E2-10n%2B25%29\"$ and factors further to $\"-4n%5E2%28n-5%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"-4n%5E4%2B40n%5E3-100n%5E2\"$ factors to $\"-4n%5E2%28n-5%29%5E2\"$ \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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