document.write( "Question 551105: Factor x^2+18x+81 \n" ); document.write( "

Algebra.Com's Answer #359388 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B18x%2B81\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"18\"$ , and the last term is $\"81\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"81\"$ to get $\"%281%29%2881%29=81\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"81\"$ (the previous product) and add to the second coefficient $\"18\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"81\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"81\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,9,27,81\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-9,-27,-81\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"81\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*81 = 81
\n" ); document.write( "3*27 = 81
\n" ); document.write( "9*9 = 81
\n" ); document.write( "(-1)*(-81) = 81
\n" ); document.write( "(-3)*(-27) = 81
\n" ); document.write( "(-9)*(-9) = 81\r
\n" ); document.write( "
\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 $\"18\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	81	1+81=82
3	27	3+27=30
9	9	9+9=18
-1	-81	-1+(-81)=-82
-3	-27	-3+(-27)=-30
-9	-9	-9+(-9)=-18

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"9\"$ and $\"9\"$ add to $\"18\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"9\"$ and $\"9\"$ both multiply to $\"81\"$ and add to $\"18\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"18x\"$ with $\"9x%2B9x\"$ . Remember, $\"9\"$ and $\"9\"$ add to $\"18\"$ . So this shows us that $\"9x%2B9x=18x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%289x%2B9x%29%2B81\"$ Replace the second term $\"18x\"$ with $\"9x%2B9x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B9x%29%2B%289x%2B81%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B9%29%2B%289x%2B81%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B9%29%2B9%28x%2B9%29\"$ 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
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B9%29%28x%2B9%29\"$ Combine like terms. Or factor out the common term $\"x%2B9\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B9%29%5E2\"$ Condense the terms.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B18x%2B81\"$ factors to $\"%28x%2B9%29%5E2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B18x%2B81=%28x%2B9%29%5E2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B9%29%5E2\"$ to get $\"x%5E2%2B18x%2B81\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );