document.write( "Question 168974: Write the coefficient of the middle term as the sum of two numbers, whose product equals the last term.
\n" ); document.write( "z^2-13z+30 \n" ); document.write( "

Algebra.Com's Answer #124599 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "Looking at the expression $\"z%5E2-13z%2B30\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"30\"$ .
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"30\"$ (the last term) and add to the second coefficient $\"-13\"$ ?\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 $\"30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"30\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,30\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30\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 $\"30\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*30
\n" ); document.write( "2*15
\n" ); document.write( "3*10
\n" ); document.write( "5*6
\n" ); document.write( "(-1)*(-30)
\n" ); document.write( "(-2)*(-15)
\n" ); document.write( "(-3)*(-10)
\n" ); document.write( "(-5)*(-6)\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 $\"-13\"$ :\r
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First Number	Second Number	Sum
1	30	1+30=31
2	15	2+15=17
3	10	3+10=13
5	6	5+6=11
-1	-30	-1+(-30)=-31
-2	-15	-2+(-15)=-17
-3	-10	-3+(-10)=-13
-5	-6	-5+(-6)=-11

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"-10\"$ add to $\"-13\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"-10\"$ both multiply to $\"30\"$ and add to $\"-13\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-13z\"$ with $\"-3z-10z\"$ . Remember, $\"-3\"$ and $\"-10\"$ add to $\"-13\"$ . So this shows us that $\"-3z-10z=-13z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"z%5E2%2Bhighlight%28-3z-10z%29%2B30\"$ Replace the second term $\"-13z\"$ with $\"-3z-10z\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So the expression breaks down into $\"z%5E2-3z-10z%2B30\"$ \n" ); document.write( "

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