document.write( "Question 960515: Hello I need some help with my math homework. I have to factor this polynomial completely. I am stuck and I need some help I have w^2(power) + 30w + 81 now I know I have to find the GCF butI dont know what to do from there, can you help me please? \n" ); document.write( "

Algebra.Com's Answer #587007 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"w%5E2%2B30w%2B81\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"30\"$ , 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 $\"30\"$ ?\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 $\"30\"$ :\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 $\"3\"$ and $\"27\"$ add to $\"30\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"27\"$ both multiply to $\"81\"$ and add to $\"30\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"30w\"$ with $\"3w%2B27w\"$ . Remember, $\"3\"$ and $\"27\"$ add to $\"30\"$ . So this shows us that $\"3w%2B27w=30w\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"w%5E2%2Bhighlight%283w%2B27w%29%2B81\"$ Replace the second term $\"30w\"$ with $\"3w%2B27w\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28w%5E2%2B3w%29%2B%2827w%2B81%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"w%28w%2B3%29%2B%2827w%2B81%29\"$ Factor out the GCF $\"w\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"w%28w%2B3%29%2B27%28w%2B3%29\"$ Factor out $\"27\"$ 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( " $\"%28w%2B27%29%28w%2B3%29\"$ Combine like terms. Or factor out the common term $\"w%2B3\"$ \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 $\"w%5E2%2B30w%2B81\"$ factors to $\"%28w%2B27%29%28w%2B3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"w%5E2%2B30w%2B81=%28w%2B27%29%28w%2B3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28w%2B27%29%28w%2B3%29\"$ to get $\"w%5E2%2B30w%2B81\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------------------------------------------------------------------\r
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Alternatively, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Any amount is greatly appreciated as it helps me a lot. This donation is to support free tutoring. Thank you.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim\r
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------------------------------------------------------------------ \n" ); document.write( "

\n" );