document.write( "Question 551103: Factor 3x^2+14x+16 \n" ); document.write( "

Algebra.Com's Answer #359374 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3x%5E2%2B14x%2B16\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"14\"$ , and the last term is $\"16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"16\"$ to get $\"%283%29%2816%29=48\"$ .\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 $\"14\"$ ?\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 = 48
\n" ); document.write( "2*24 = 48
\n" ); document.write( "3*16 = 48
\n" ); document.write( "4*12 = 48
\n" ); document.write( "6*8 = 48
\n" ); document.write( "(-1)*(-48) = 48
\n" ); document.write( "(-2)*(-24) = 48
\n" ); document.write( "(-3)*(-16) = 48
\n" ); document.write( "(-4)*(-12) = 48
\n" ); document.write( "(-6)*(-8) = 48\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 $\"14\"$ :\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 $\"6\"$ and $\"8\"$ add to $\"14\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"6\"$ and $\"8\"$ both multiply to $\"48\"$ and add to $\"14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"14x\"$ with $\"6x%2B8x\"$ . Remember, $\"6\"$ and $\"8\"$ add to $\"14\"$ . So this shows us that $\"6x%2B8x=14x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%286x%2B8x%29%2B16\"$ Replace the second term $\"14x\"$ with $\"6x%2B8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2%2B6x%29%2B%288x%2B16%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%28x%2B2%29%2B%288x%2B16%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%28x%2B2%29%2B8%28x%2B2%29\"$ 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( " $\"%283x%2B8%29%28x%2B2%29\"$ Combine like terms. Or factor out the common term $\"x%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B14x%2B16\"$ factors to $\"%283x%2B8%29%28x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2%2B14x%2B16=%283x%2B8%29%28x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283x%2B8%29%28x%2B2%29\"$ to get $\"3x%5E2%2B14x%2B16\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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