document.write( "Question 778412: Factor by using trial factors.\r
\n" ); document.write( "\n" ); document.write( "3p^2 + 20p - 32 \n" ); document.write( "

Algebra.Com's Answer #474606 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3p%5E2%2B20p-32\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"20\"$ , and the last term is $\"-32\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"-32\"$ to get $\"%283%29%28-32%29=-96\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-96\"$ (the previous product) and add to the second coefficient $\"20\"$ ?\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 $\"-96\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-96\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,32,48,96\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96\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 $\"-96\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-96) = -96
\n" ); document.write( "2*(-48) = -96
\n" ); document.write( "3*(-32) = -96
\n" ); document.write( "4*(-24) = -96
\n" ); document.write( "6*(-16) = -96
\n" ); document.write( "8*(-12) = -96
\n" ); document.write( "(-1)*(96) = -96
\n" ); document.write( "(-2)*(48) = -96
\n" ); document.write( "(-3)*(32) = -96
\n" ); document.write( "(-4)*(24) = -96
\n" ); document.write( "(-6)*(16) = -96
\n" ); document.write( "(-8)*(12) = -96\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 $\"20\"$ :\r
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First Number	Second Number	Sum
1	-96	1+(-96)=-95
2	-48	2+(-48)=-46
3	-32	3+(-32)=-29
4	-24	4+(-24)=-20
6	-16	6+(-16)=-10
8	-12	8+(-12)=-4
-1	96	-1+96=95
-2	48	-2+48=46
-3	32	-3+32=29
-4	24	-4+24=20
-6	16	-6+16=10
-8	12	-8+12=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"24\"$ add to $\"20\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"24\"$ both multiply to $\"-96\"$ and add to $\"20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"20p\"$ with $\"-4p%2B24p\"$ . Remember, $\"-4\"$ and $\"24\"$ add to $\"20\"$ . So this shows us that $\"-4p%2B24p=20p\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3p%5E2%2Bhighlight%28-4p%2B24p%29-32\"$ Replace the second term $\"20p\"$ with $\"-4p%2B24p\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283p%5E2-4p%29%2B%2824p-32%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"p%283p-4%29%2B%2824p-32%29\"$ Factor out the GCF $\"p\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"p%283p-4%29%2B8%283p-4%29\"$ Factor out $\"8\"$ 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( " $\"%28p%2B8%29%283p-4%29\"$ Combine like terms. Or factor out the common term $\"3p-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3p%5E2%2B20p-32\"$ factors to $\"%28p%2B8%29%283p-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3p%5E2%2B20p-32=%28p%2B8%29%283p-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28p%2B8%29%283p-4%29\"$ to get $\"3p%5E2%2B20p-32\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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