document.write( "Question 160428: Factor using ac method
\n" ); document.write( "4z^2 - 5z - 6 \n" ); document.write( "

Algebra.Com's Answer #118337 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4z%5E2-5z-6\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"-6\"$ to get $\"%284%29%28-6%29=-24\"$ .\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 $\"-5\"$ ?\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)
\n" ); document.write( "2*(-12)
\n" ); document.write( "3*(-8)
\n" ); document.write( "4*(-6)
\n" ); document.write( "(-1)*(24)
\n" ); document.write( "(-2)*(12)
\n" ); document.write( "(-3)*(8)
\n" ); document.write( "(-4)*(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 $\"-5\"$ :\r
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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"-8\"$ add to $\"-5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-8\"$ both multiply to $\"-24\"$ and add to $\"-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5z\"$ with $\"3z-8z\"$ . Remember, $\"3\"$ and $\"-8\"$ add to $\"-5\"$ . So this shows us that $\"3z-8z=-5z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4z%5E2%2Bhighlight%283z-8z%29-6\"$ Replace the second term $\"-5z\"$ with $\"3z-8z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284z%5E2%2B3z%29%2B%28-8z-6%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%284z%2B3%29%2B%28-8z-6%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%284z%2B3%29-2%284z%2B3%29\"$ 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( " $\"%28z-2%29%284z%2B3%29\"$ Combine like terms. Or factor out the common term $\"4z%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4z%5E2-5z-6\"$ factors to $\"%28z-2%29%284z%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28z-2%29%284z%2B3%29\"$ to get $\"4z%5E2-5z-6\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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