document.write( "Question 610228: Factor completely. If a polynomial is prime, state this.
\n" ); document.write( "-12x^2-28x+24 \n" ); document.write( "

Algebra.Com's Answer #384226 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"-12x%5E2-28x%2B24\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"-4%283x%5E2%2B7x-6%29\"$ Factor out the GCF $\"-4\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"3x%5E2%2B7x-6\"$ \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( "Looking at the expression $\"3x%5E2%2B7x-6\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-6\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"-6\"$ to get $\"%283%29%28-6%29=-18\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-18\"$ (the previous product) and add to the second coefficient $\"7\"$ ?\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 $\"-18\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-18\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,9,18\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-9,-18\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 $\"-18\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-18) = -18
\n" ); document.write( "2*(-9) = -18
\n" ); document.write( "3*(-6) = -18
\n" ); document.write( "(-1)*(18) = -18
\n" ); document.write( "(-2)*(9) = -18
\n" ); document.write( "(-3)*(6) = -18\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 $\"7\"$ :\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	-18	1+(-18)=-17
2	-9	2+(-9)=-7
3	-6	3+(-6)=-3
-1	18	-1+18=17
-2	9	-2+9=7
-3	6	-3+6=3

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"9\"$ add to $\"7\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"9\"$ both multiply to $\"-18\"$ and add to $\"7\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"7x\"$ with $\"-2x%2B9x\"$ . Remember, $\"-2\"$ and $\"9\"$ add to $\"7\"$ . So this shows us that $\"-2x%2B9x=7x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-2x%2B9x%29-6\"$ Replace the second term $\"7x\"$ with $\"-2x%2B9x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-2x%29%2B%289x-6%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%283x-2%29%2B%289x-6%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%283x-2%29%2B3%283x-2%29\"$ Factor out $\"3\"$ 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( " $\"%28x%2B3%29%283x-2%29\"$ Combine like terms. Or factor out the common term $\"3x-2\"$ \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( "So $\"-4%283x%5E2%2B7x-6%29\"$ then factors further to $\"-4%28x%2B3%29%283x-2%29\"$ \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 $\"-12x%5E2-28x%2B24\"$ completely factors to $\"-4%28x%2B3%29%283x-2%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"-12x%5E2-28x%2B24=-4%28x%2B3%29%283x-2%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"-4%28x%2B3%29%283x-2%29\"$ to get $\"-12x%5E2-28x%2B24\"$ 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( "--------------------------------------------------------------------------------------------------------------
\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim
\n" ); document.write( "-------------------------------------------------------------------------------------------------------------- \n" ); document.write( "

\n" );