document.write( "Question 277689: factor z^2-10z+24 \n" ); document.write( "

Algebra.Com's Answer #202139 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"z%5E2-10z%2B24\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-10\"$ , and the last term is $\"24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"24\"$ to get $\"%281%29%2824%29=24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"24\"$ (the previous product) and add to the second coefficient $\"-10\"$ ?\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 $\"24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"24\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\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 $\"24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*24 = 24
\n" ); document.write( "2*12 = 24
\n" ); document.write( "3*8 = 24
\n" ); document.write( "4*6 = 24
\n" ); document.write( "(-1)*(-24) = 24
\n" ); document.write( "(-2)*(-12) = 24
\n" ); document.write( "(-3)*(-8) = 24
\n" ); document.write( "(-4)*(-6) = 24\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 $\"-10\"$ :\r
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First Number	Second Number	Sum
1	24	1+24=25
2	12	2+12=14
3	8	3+8=11
4	6	4+6=10
-1	-24	-1+(-24)=-25
-2	-12	-2+(-12)=-14
-3	-8	-3+(-8)=-11
-4	-6	-4+(-6)=-10

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"-6\"$ add to $\"-10\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"-6\"$ both multiply to $\"24\"$ and add to $\"-10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-10z\"$ with $\"-4z-6z\"$ . Remember, $\"-4\"$ and $\"-6\"$ add to $\"-10\"$ . So this shows us that $\"-4z-6z=-10z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"z%5E2%2Bhighlight%28-4z-6z%29%2B24\"$ Replace the second term $\"-10z\"$ with $\"-4z-6z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28z%5E2-4z%29%2B%28-6z%2B24%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z-4%29%2B%28-6z%2B24%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z-4%29-6%28z-4%29\"$ 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( " $\"%28z-6%29%28z-4%29\"$ Combine like terms. Or factor out the common term $\"z-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"z%5E2-10z%2B24\"$ factors to $\"%28z-6%29%28z-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"z%5E2-10z%2B24=%28z-6%29%28z-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28z-6%29%28z-4%29\"$ to get $\"z%5E2-10z%2B24\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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