document.write( "Question 204682This question is from textbook
\n" ); document.write( ": I need help factoring 12x squared+19x+4 \n" ); document.write( "

Algebra.Com's Answer #154479 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12x%5E2%2B19x%2B4\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"19\"$ , and the last term is $\"4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last term $\"4\"$ to get $\"%2812%29%284%29=48\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"48\"$ (the previous product) and add to the second coefficient $\"19\"$ ?\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 $\"48\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"48\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,48\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48\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 $\"48\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*48
\n" ); document.write( "2*24
\n" ); document.write( "3*16
\n" ); document.write( "4*12
\n" ); document.write( "6*8
\n" ); document.write( "(-1)*(-48)
\n" ); document.write( "(-2)*(-24)
\n" ); document.write( "(-3)*(-16)
\n" ); document.write( "(-4)*(-12)
\n" ); document.write( "(-6)*(-8)\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 $\"19\"$ :\r
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First Number	Second Number	Sum
1	48	1+48=49
2	24	2+24=26
3	16	3+16=19
4	12	4+12=16
6	8	6+8=14
-1	-48	-1+(-48)=-49
-2	-24	-2+(-24)=-26
-3	-16	-3+(-16)=-19
-4	-12	-4+(-12)=-16
-6	-8	-6+(-8)=-14

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"16\"$ add to $\"19\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"16\"$ both multiply to $\"48\"$ and add to $\"19\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"19x\"$ with $\"3x%2B16x\"$ . Remember, $\"3\"$ and $\"16\"$ add to $\"19\"$ . So this shows us that $\"3x%2B16x=19x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2%2Bhighlight%283x%2B16x%29%2B4\"$ Replace the second term $\"19x\"$ with $\"3x%2B16x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812x%5E2%2B3x%29%2B%2816x%2B4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2B1%29%2B%2816x%2B4%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2B1%29%2B4%284x%2B1%29\"$ Factor out $\"4\"$ 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( " $\"%283x%2B4%29%284x%2B1%29\"$ Combine like terms. Or factor out the common term $\"4x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E2%2B19x%2B4\"$ factors to $\"%283x%2B4%29%284x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%283x%2B4%29%284x%2B1%29\"$ to get $\"12x%5E2%2B19x%2B4\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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