document.write( "Question 742464: 5z^2+19z-4 \n" ); document.write( "

Algebra.Com's Answer #452484 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5z%5E2%2B19z-4\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"19\"$ , and the last term is $\"-4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"-4\"$ to get $\"%285%29%28-4%29=-20\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-20\"$ (the previous product) and add to the second coefficient $\"19\"$ ?\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 $\"-20\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-20\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,10,20\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20\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 $\"-20\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-20) = -20
\n" ); document.write( "2*(-10) = -20
\n" ); document.write( "4*(-5) = -20
\n" ); document.write( "(-1)*(20) = -20
\n" ); document.write( "(-2)*(10) = -20
\n" ); document.write( "(-4)*(5) = -20\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 $\"19\"$ :\r
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First Number	Second Number	Sum
1	-20	1+(-20)=-19
2	-10	2+(-10)=-8
4	-5	4+(-5)=-1
-1	20	-1+20=19
-2	10	-2+10=8
-4	5	-4+5=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"20\"$ add to $\"19\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"20\"$ both multiply to $\"-20\"$ and add to $\"19\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"19z\"$ with $\"-z%2B20z\"$ . Remember, $\"-1\"$ and $\"20\"$ add to $\"19\"$ . So this shows us that $\"-z%2B20z=19z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5z%5E2%2Bhighlight%28-z%2B20z%29-4\"$ Replace the second term $\"19z\"$ with $\"-z%2B20z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285z%5E2-z%29%2B%2820z-4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%285z-1%29%2B%2820z-4%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%285z-1%29%2B4%285z-1%29\"$ Factor out $\"4\"$ 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( " $\"%28z%2B4%29%285z-1%29\"$ Combine like terms. Or factor out the common term $\"5z-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5z%5E2%2B19z-4\"$ factors to $\"%28z%2B4%29%285z-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"5z%5E2%2B19z-4=%28z%2B4%29%285z-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28z%2B4%29%285z-1%29\"$ to get $\"5z%5E2%2B19z-4\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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