document.write( "Question 185528: please help with this polynomial
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\n" ); document.write( "10x^2+23x+12
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Algebra.Com's Answer #139180 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I assume that you want to factor.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"10x%5E2%2B23x%2B12\"$ , we can see that the first coefficient is $\"10\"$ , the second coefficient is $\"23\"$ , and the last term is $\"12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"10\"$ by the last term $\"12\"$ to get $\"%2810%29%2812%29=120\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"120\"$ (the previous product) and add to the second coefficient $\"23\"$ ?\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 $\"120\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"120\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120\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 $\"120\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*120
\n" ); document.write( "2*60
\n" ); document.write( "3*40
\n" ); document.write( "4*30
\n" ); document.write( "5*24
\n" ); document.write( "6*20
\n" ); document.write( "8*15
\n" ); document.write( "10*12
\n" ); document.write( "(-1)*(-120)
\n" ); document.write( "(-2)*(-60)
\n" ); document.write( "(-3)*(-40)
\n" ); document.write( "(-4)*(-30)
\n" ); document.write( "(-5)*(-24)
\n" ); document.write( "(-6)*(-20)
\n" ); document.write( "(-8)*(-15)
\n" ); document.write( "(-10)*(-12)\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 $\"23\"$ :\r
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First Number	Second Number	Sum
1	120	1+120=121
2	60	2+60=62
3	40	3+40=43
4	30	4+30=34
5	24	5+24=29
6	20	6+20=26
8	15	8+15=23
10	12	10+12=22
-1	-120	-1+(-120)=-121
-2	-60	-2+(-60)=-62
-3	-40	-3+(-40)=-43
-4	-30	-4+(-30)=-34
-5	-24	-5+(-24)=-29
-6	-20	-6+(-20)=-26
-8	-15	-8+(-15)=-23
-10	-12	-10+(-12)=-22

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"8\"$ and $\"15\"$ add to $\"23\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"8\"$ and $\"15\"$ both multiply to $\"120\"$ and add to $\"23\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"23x\"$ with $\"8x%2B15x\"$ . Remember, $\"8\"$ and $\"15\"$ add to $\"23\"$ . So this shows us that $\"8x%2B15x=23x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"10x%5E2%2Bhighlight%288x%2B15x%29%2B12\"$ Replace the second term $\"23x\"$ with $\"8x%2B15x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2810x%5E2%2B8x%29%2B%2815x%2B12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%285x%2B4%29%2B%2815x%2B12%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%285x%2B4%29%2B3%285x%2B4%29\"$ Factor out $\"3\"$ 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( " $\"%282x%2B3%29%285x%2B4%29\"$ Combine like terms. Or factor out the common term $\"5x%2B4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"10x%5E2%2B23x%2B12\"$ factors to $\"%282x%2B3%29%285x%2B4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282x%2B3%29%285x%2B4%29\"$ to get $\"10x%5E2%2B23x%2B12\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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