document.write( "Question 593881: factor z^2+8z+12 \n" ); document.write( "

Algebra.Com's Answer #376508 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"z%5E2%2B8z%2B12\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"8\"$ , and the last term is $\"12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"12\"$ to get $\"%281%29%2812%29=12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"12\"$ (the previous product) and add to the second coefficient $\"8\"$ ?\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 $\"12\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"12\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12\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 $\"12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*12 = 12
\n" ); document.write( "2*6 = 12
\n" ); document.write( "3*4 = 12
\n" ); document.write( "(-1)*(-12) = 12
\n" ); document.write( "(-2)*(-6) = 12
\n" ); document.write( "(-3)*(-4) = 12\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 $\"8\"$ :\r
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First Number	Second Number	Sum
1	12	1+12=13
2	6	2+6=8
3	4	3+4=7
-1	-12	-1+(-12)=-13
-2	-6	-2+(-6)=-8
-3	-4	-3+(-4)=-7

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"6\"$ add to $\"8\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"6\"$ both multiply to $\"12\"$ and add to $\"8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"8z\"$ with $\"2z%2B6z\"$ . Remember, $\"2\"$ and $\"6\"$ add to $\"8\"$ . So this shows us that $\"2z%2B6z=8z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"z%5E2%2Bhighlight%282z%2B6z%29%2B12\"$ Replace the second term $\"8z\"$ with $\"2z%2B6z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28z%5E2%2B2z%29%2B%286z%2B12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z%2B2%29%2B%286z%2B12%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z%2B2%29%2B6%28z%2B2%29\"$ Factor out $\"6\"$ 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( " $\"%28z%2B6%29%28z%2B2%29\"$ Combine like terms. Or factor out the common term $\"z%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"z%5E2%2B8z%2B12\"$ factors to $\"%28z%2B6%29%28z%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"z%5E2%2B8z%2B12=%28z%2B6%29%28z%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28z%2B6%29%28z%2B2%29\"$ to get $\"z%5E2%2B8z%2B12\"$ 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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