document.write( "Question 552838: Factor Completely\r
\n" ); document.write( "\n" ); document.write( "81s^2+16-72s \n" ); document.write( "

Algebra.Com's Answer #360527 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"81s%5E2-72s%2B16\"$ , we can see that the first coefficient is $\"81\"$ , the second coefficient is $\"-72\"$ , and the last term is $\"16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"81\"$ by the last term $\"16\"$ to get $\"%2881%29%2816%29=1296\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"1296\"$ (the previous product) and add to the second coefficient $\"-72\"$ ?\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 $\"1296\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"1296\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,27,36,48,54,72,81,108,144,162,216,324,432,648,1296\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-27,-36,-48,-54,-72,-81,-108,-144,-162,-216,-324,-432,-648,-1296\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 $\"1296\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*1296 = 1296
\n" ); document.write( "2*648 = 1296
\n" ); document.write( "3*432 = 1296
\n" ); document.write( "4*324 = 1296
\n" ); document.write( "6*216 = 1296
\n" ); document.write( "8*162 = 1296
\n" ); document.write( "9*144 = 1296
\n" ); document.write( "12*108 = 1296
\n" ); document.write( "16*81 = 1296
\n" ); document.write( "18*72 = 1296
\n" ); document.write( "24*54 = 1296
\n" ); document.write( "27*48 = 1296
\n" ); document.write( "36*36 = 1296
\n" ); document.write( "(-1)*(-1296) = 1296
\n" ); document.write( "(-2)*(-648) = 1296
\n" ); document.write( "(-3)*(-432) = 1296
\n" ); document.write( "(-4)*(-324) = 1296
\n" ); document.write( "(-6)*(-216) = 1296
\n" ); document.write( "(-8)*(-162) = 1296
\n" ); document.write( "(-9)*(-144) = 1296
\n" ); document.write( "(-12)*(-108) = 1296
\n" ); document.write( "(-16)*(-81) = 1296
\n" ); document.write( "(-18)*(-72) = 1296
\n" ); document.write( "(-24)*(-54) = 1296
\n" ); document.write( "(-27)*(-48) = 1296
\n" ); document.write( "(-36)*(-36) = 1296\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 $\"-72\"$ :\r
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First Number	Second Number	Sum
1	1296	1+1296=1297
2	648	2+648=650
3	432	3+432=435
4	324	4+324=328
6	216	6+216=222
8	162	8+162=170
9	144	9+144=153
12	108	12+108=120
16	81	16+81=97
18	72	18+72=90
24	54	24+54=78
27	48	27+48=75
36	36	36+36=72
-1	-1296	-1+(-1296)=-1297
-2	-648	-2+(-648)=-650
-3	-432	-3+(-432)=-435
-4	-324	-4+(-324)=-328
-6	-216	-6+(-216)=-222
-8	-162	-8+(-162)=-170
-9	-144	-9+(-144)=-153
-12	-108	-12+(-108)=-120
-16	-81	-16+(-81)=-97
-18	-72	-18+(-72)=-90
-24	-54	-24+(-54)=-78
-27	-48	-27+(-48)=-75
-36	-36	-36+(-36)=-72

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-36\"$ and $\"-36\"$ add to $\"-72\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-36\"$ and $\"-36\"$ both multiply to $\"1296\"$ and add to $\"-72\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-72s\"$ with $\"-36s-36s\"$ . Remember, $\"-36\"$ and $\"-36\"$ add to $\"-72\"$ . So this shows us that $\"-36s-36s=-72s\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"81s%5E2%2Bhighlight%28-36s-36s%29%2B16\"$ Replace the second term $\"-72s\"$ with $\"-36s-36s\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2881s%5E2-36s%29%2B%28-36s%2B16%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"9s%289s-4%29%2B%28-36s%2B16%29\"$ Factor out the GCF $\"9s\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"9s%289s-4%29-4%289s-4%29\"$ 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( " $\"%289s-4%29%289s-4%29\"$ Combine like terms. Or factor out the common term $\"9s-4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%289s-4%29%5E2\"$ 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 $\"81s%5E2-72s%2B16\"$ factors to $\"%289s-4%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"81s%5E2-72s%2B16=%289s-4%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%289s-4%29%5E2\"$ to get $\"81s%5E2-72s%2B16\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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