document.write( "Question 960518: How do you factor this special case polynomial?\r
\n" ); document.write( "\n" ); document.write( "4d squared + 36d + 81 \n" ); document.write( "

Algebra.Com's Answer #587012 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4d%5E2%2B36d%2B81\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"36\"$ , and the last term is $\"81\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"81\"$ to get $\"%284%29%2881%29=324\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"324\"$ (the previous product) and add to the second coefficient $\"36\"$ ?\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 $\"324\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"324\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,27,36,54,81,108,162,324\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162,-324\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 $\"324\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*324 = 324
\n" ); document.write( "2*162 = 324
\n" ); document.write( "3*108 = 324
\n" ); document.write( "4*81 = 324
\n" ); document.write( "6*54 = 324
\n" ); document.write( "9*36 = 324
\n" ); document.write( "12*27 = 324
\n" ); document.write( "18*18 = 324
\n" ); document.write( "(-1)*(-324) = 324
\n" ); document.write( "(-2)*(-162) = 324
\n" ); document.write( "(-3)*(-108) = 324
\n" ); document.write( "(-4)*(-81) = 324
\n" ); document.write( "(-6)*(-54) = 324
\n" ); document.write( "(-9)*(-36) = 324
\n" ); document.write( "(-12)*(-27) = 324
\n" ); document.write( "(-18)*(-18) = 324\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 $\"36\"$ :\r
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First Number	Second Number	Sum
1	324	1+324=325
2	162	2+162=164
3	108	3+108=111
4	81	4+81=85
6	54	6+54=60
9	36	9+36=45
12	27	12+27=39
18	18	18+18=36
-1	-324	-1+(-324)=-325
-2	-162	-2+(-162)=-164
-3	-108	-3+(-108)=-111
-4	-81	-4+(-81)=-85
-6	-54	-6+(-54)=-60
-9	-36	-9+(-36)=-45
-12	-27	-12+(-27)=-39
-18	-18	-18+(-18)=-36

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"18\"$ and $\"18\"$ add to $\"36\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"18\"$ and $\"18\"$ both multiply to $\"324\"$ and add to $\"36\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"36d\"$ with $\"18d%2B18d\"$ . Remember, $\"18\"$ and $\"18\"$ add to $\"36\"$ . So this shows us that $\"18d%2B18d=36d\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4d%5E2%2Bhighlight%2818d%2B18d%29%2B81\"$ Replace the second term $\"36d\"$ with $\"18d%2B18d\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284d%5E2%2B18d%29%2B%2818d%2B81%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2d%282d%2B9%29%2B%2818d%2B81%29\"$ Factor out the GCF $\"2d\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2d%282d%2B9%29%2B9%282d%2B9%29\"$ Factor out $\"9\"$ 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( " $\"%282d%2B9%29%282d%2B9%29\"$ Combine like terms. Or factor out the common term $\"2d%2B9\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%282d%2B9%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4d%5E2%2B36d%2B81\"$ factors to $\"%282d%2B9%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"4d%5E2%2B36d%2B81=%282d%2B9%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282d%2B9%29%5E2\"$ to get $\"4d%5E2%2B36d%2B81\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more one-on-one help, email me at jim_thompson5910@hotmail.com. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).\r
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\n" ); document.write( "\n" ); document.write( "Jim\r
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