document.write( "Question 790014: 5z^2-17z+14 \n" ); document.write( "

Algebra.Com's Answer #479011 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5z%5E2-17z%2B14\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"-17\"$ , and the last term is $\"14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"14\"$ to get $\"%285%29%2814%29=70\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"70\"$ (the previous product) and add to the second coefficient $\"-17\"$ ?\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 $\"70\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"70\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,5,7,10,14,35,70\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-7,-10,-14,-35,-70\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 $\"70\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*70 = 70
\n" ); document.write( "2*35 = 70
\n" ); document.write( "5*14 = 70
\n" ); document.write( "7*10 = 70
\n" ); document.write( "(-1)*(-70) = 70
\n" ); document.write( "(-2)*(-35) = 70
\n" ); document.write( "(-5)*(-14) = 70
\n" ); document.write( "(-7)*(-10) = 70\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 $\"-17\"$ :\r
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First Number	Second Number	Sum
1	70	1+70=71
2	35	2+35=37
5	14	5+14=19
7	10	7+10=17
-1	-70	-1+(-70)=-71
-2	-35	-2+(-35)=-37
-5	-14	-5+(-14)=-19
-7	-10	-7+(-10)=-17

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-7\"$ and $\"-10\"$ add to $\"-17\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-7\"$ and $\"-10\"$ both multiply to $\"70\"$ and add to $\"-17\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-17z\"$ with $\"-7z-10z\"$ . Remember, $\"-7\"$ and $\"-10\"$ add to $\"-17\"$ . So this shows us that $\"-7z-10z=-17z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5z%5E2%2Bhighlight%28-7z-10z%29%2B14\"$ Replace the second term $\"-17z\"$ with $\"-7z-10z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285z%5E2-7z%29%2B%28-10z%2B14%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%285z-7%29%2B%28-10z%2B14%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%285z-7%29-2%285z-7%29\"$ Factor out $\"2\"$ 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( " $\"%28z-2%29%285z-7%29\"$ Combine like terms. Or factor out the common term $\"5z-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5z%5E2-17z%2B14\"$ factors to $\"%28z-2%29%285z-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"5z%5E2-17z%2B14=%28z-2%29%285z-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28z-2%29%285z-7%29\"$ to get $\"5z%5E2-17z%2B14\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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