document.write( "Question 652174: Factor the trinominal \r
\n" ); document.write( "\n" ); document.write( "y^4+9y^3+8y^2\r
\n" ); document.write( "\n" ); document.write( "(y+8)(y+1) \n" ); document.write( "

Algebra.Com's Answer #407922 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"y%5E4%2B9y%5E3%2B8y%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%28y%5E2%2B9y%2B8%29\"$ Factor out the GCF $\"y%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"y%5E2%2B9y%2B8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"y%5E2%2B9y%2B8\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"9\"$ , and the last term is $\"8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"8\"$ to get $\"%281%29%288%29=8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"8\"$ (the previous product) and add to the second coefficient $\"9\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"8\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"8\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"8\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*8 = 8
\n" ); document.write( "2*4 = 8
\n" ); document.write( "(-1)*(-8) = 8
\n" ); document.write( "(-2)*(-4) = 8\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"9\"$ :\r
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First Number	Second Number	Sum
1	8	1+8=9
2	4	2+4=6
-1	-8	-1+(-8)=-9
-2	-4	-2+(-4)=-6

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"1\"$ and $\"8\"$ add to $\"9\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"1\"$ and $\"8\"$ both multiply to $\"8\"$ and add to $\"9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"9y\"$ with $\"y%2B8y\"$ . Remember, $\"1\"$ and $\"8\"$ add to $\"9\"$ . So this shows us that $\"y%2B8y=9y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2Bhighlight%28y%2B8y%29%2B8\"$ Replace the second term $\"9y\"$ with $\"y%2B8y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%5E2%2By%29%2B%288y%2B8%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y%2B1%29%2B%288y%2B8%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y%2B1%29%2B8%28y%2B1%29\"$ Factor out $\"8\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%2B8%29%28y%2B1%29\"$ Combine like terms. Or factor out the common term $\"y%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"y%5E2%28y%5E2%2B9y%2B8%29\"$ then factors further to $\"y%5E2%28y%2B8%29%28y%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"y%5E4%2B9y%5E3%2B8y%5E2\"$ completely factors to $\"y%5E2%28y%2B8%29%28y%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"y%5E4%2B9y%5E3%2B8y%5E2=y%5E2%28y%2B8%29%28y%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"y%5E2%28y%2B8%29%28y%2B1%29\"$ to get $\"y%5E4%2B9y%5E3%2B8y%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\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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