document.write( "Question 131050: when z^2+5z-84 is factored completely, one of the factors is: \n" ); document.write( "

Algebra.Com's Answer #95649 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"1z%5E2%2B5z-84\"$ we can see that the first term is $\"1z%5E2\"$ and the last term is $\"-84\"$ where the coefficients are 1 and -84 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -84 to get -84. Now what two numbers multiply to -84 and add to the middle coefficient 5? Let's list all of the factors of -84:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -84:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,12,14,21,28,42,84\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84 ...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 -84\r
\n" ); document.write( "\n" ); document.write( "(1)*(-84)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-42)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-28)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-21)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-14)\r
\n" ); document.write( "\n" ); document.write( "(7)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(84)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(42)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(28)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(21)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(14)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(12)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5\r
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First Number	Second Number	Sum
1	-84	1+(-84)=-83
2	-42	2+(-42)=-40
3	-28	3+(-28)=-25
4	-21	4+(-21)=-17
6	-14	6+(-14)=-8
7	-12	7+(-12)=-5
-1	84	-1+84=83
-2	42	-2+42=40
-3	28	-3+28=25
-4	21	-4+21=17
-6	14	-6+14=8
-7	12	-7+12=5

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -7 and 12 add up to 5 and multiply to -84\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"1z%5E2%2B5z-84\"$ , replace $\"5z\"$ with $\"-7z%2B12z\"$ (notice $\"-7z%2B12z\"$ adds up to $\"5z\"$ . So it is equivalent to $\"5z\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"1z%5E2%2Bhighlight%28-7z%2B12z%29%2B-84\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1z%5E2-7z%2B12z-84\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%281z%5E2-7z%29%2B%2812z-84%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z-7%29%2B12%28z-7%29\"$ Factor out the GCF of $\"z\"$ out of the first group. Factor out the GCF of $\"12\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28z%2B12%29%28z-7%29\"$ Since we have a common term of $\"z-7\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"1z%5E2-7z%2B12z-84\"$ factors to $\"%28z%2B12%29%28z-7%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"1z%5E2%2B5z-84\"$ factors to $\"%28z%2B12%29%28z-7%29\"$ (since $\"1z%5E2%2B5z-84\"$ is equivalent to $\"1z%5E2-7z%2B12z-84\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"z%5E2%2B5z-84\"$ factors to $\"%28z%2B12%29%28z-7%29\"$ which means that one of the factors are either $\"z%2B12\"$ or $\"z-7\"$ \n" ); document.write( "

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