document.write( "Question 629843: Factor the trinomial by grouping\r
\n" ); document.write( "\n" ); document.write( "4x^3-10x^2-50x \n" ); document.write( "

Algebra.Com's Answer #396552 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"4x%5E3-10x%5E2-50x\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%282x%5E2-5x-25%29\"$ Factor out the GCF $\"2x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"2x%5E2-5x-25\"$ \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 $\"2x%5E2-5x-25\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-25\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"-25\"$ to get $\"%282%29%28-25%29=-50\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-50\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\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 $\"-50\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-50\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,5,10,25,50\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-10,-25,-50\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 $\"-50\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-50) = -50
\n" ); document.write( "2*(-25) = -50
\n" ); document.write( "5*(-10) = -50
\n" ); document.write( "(-1)*(50) = -50
\n" ); document.write( "(-2)*(25) = -50
\n" ); document.write( "(-5)*(10) = -50\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 $\"-5\"$ :\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	-50	1+(-50)=-49
2	-25	2+(-25)=-23
5	-10	5+(-10)=-5
-1	50	-1+50=49
-2	25	-2+25=23
-5	10	-5+10=5

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"-10\"$ add to $\"-5\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"-10\"$ both multiply to $\"-50\"$ and add to $\"-5\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5x\"$ with $\"5x-10x\"$ . Remember, $\"5\"$ and $\"-10\"$ add to $\"-5\"$ . So this shows us that $\"5x-10x=-5x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%285x-10x%29-25\"$ Replace the second term $\"-5x\"$ with $\"5x-10x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2%2B5x%29%2B%28-10x-25%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B5%29%2B%28-10x-25%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B5%29-5%282x%2B5%29\"$ Factor out $\"5\"$ 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( " $\"%28x-5%29%282x%2B5%29\"$ Combine like terms. Or factor out the common term $\"2x%2B5\"$ \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 $\"2x%282x%5E2-5x-25%29\"$ then factors further to $\"2x%28x-5%29%282x%2B5%29\"$ \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 $\"4x%5E3-10x%5E2-50x\"$ completely factors to $\"2x%28x-5%29%282x%2B5%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"4x%5E3-10x%5E2-50x=2x%28x-5%29%282x%2B5%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2x%28x-5%29%282x%2B5%29\"$ to get $\"4x%5E3-10x%5E2-50x\"$ 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" );