document.write( "Question 146140This question is from textbook
\n" ); document.write( ": FACTORING: 16y^2 + 24y + 9 there is no GCF of any of these numbers
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Algebra.Com's Answer #106689 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"16y%5E2%2B24y%2B9\"$ we can see that the first term is $\"16y%5E2\"$ and the last term is $\"9\"$ where the coefficients are 16 and 9 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 16 and the last coefficient 9 to get 144. Now what two numbers multiply to 144 and add to the middle coefficient 24? Let's list all of the factors of 144:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 144:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,36,48,72\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. 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 144\r
\n" ); document.write( "\n" ); document.write( "1*144\r
\n" ); document.write( "\n" ); document.write( "2*72\r
\n" ); document.write( "\n" ); document.write( "3*48\r
\n" ); document.write( "\n" ); document.write( "4*36\r
\n" ); document.write( "\n" ); document.write( "6*24\r
\n" ); document.write( "\n" ); document.write( "8*18\r
\n" ); document.write( "\n" ); document.write( "9*16\r
\n" ); document.write( "\n" ); document.write( "12*12\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-144)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-72)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(-12)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 24? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 24\r
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First Number	Second Number	Sum
1	144	1+144=145
2	72	2+72=74
3	48	3+48=51
4	36	4+36=40
6	24	6+24=30
8	18	8+18=26
9	16	9+16=25
12	12	12+12=24
-1	-144	-1+(-144)=-145
-2	-72	-2+(-72)=-74
-3	-48	-3+(-48)=-51
-4	-36	-4+(-36)=-40
-6	-24	-6+(-24)=-30
-8	-18	-8+(-18)=-26
-9	-16	-9+(-16)=-25
-12	-12	-12+(-12)=-24

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 12 and 12 add up to 24 and multiply to 144\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"16y%5E2%2B24y%2B9\"$ , replace $\"24y\"$ with $\"12y%2B12y\"$ (notice $\"12y%2B12y\"$ adds up to $\"24y\"$ . So it is equivalent to $\"24y\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"16y%5E2%2Bhighlight%2812y%2B12y%29%2B9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"16y%5E2%2B12y%2B12y%2B9\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2816y%5E2%2B12y%29%2B%2812y%2B9%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"4y%284y%2B3%29%2B3%284y%2B3%29\"$ Factor out the GCF of $\"4y\"$ out of the first group. Factor out the GCF of $\"3\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%284y%2B3%29%284y%2B3%29\"$ Since we have a common term of $\"4y%2B3\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"16y%5E2%2B12y%2B12y%2B9\"$ factors to $\"%284y%2B3%29%284y%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"16y%5E2%2B24y%2B9\"$ factors to $\"%284y%2B3%29%284y%2B3%29\"$ (since $\"16y%5E2%2B24y%2B9\"$ is equivalent to $\"16y%5E2%2B12y%2B12y%2B9\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%284y%2B3%29%284y%2B3%29\"$ is equivalent to $\"%284y%2B3%29%5E2\"$ since the term $\"4y%2B3\"$ occurs twice. So $\"16y%5E2%2B24y%2B9\"$ also factors to $\"%284y%2B3%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"16y%5E2%2B24y%2B9\"$ factors to $\"%284y%2B3%29%5E2\"$
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