document.write( "Question 200296: Factor: 8x^2-16-28x \n" ); document.write( "

Algebra.Com's Answer #150577 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"8x%5E2-16-28x\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"8x%5E2-28x-16\"$ Rearrange the terms.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4%282x%5E2-7x-4%29\"$ Factor out the GCF $\"4\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"2x%5E2-7x-4\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\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( "\n" ); document.write( "Looking at $\"2x%5E2-7x-4\"$ we can see that the first term is $\"2x%5E2\"$ and the last term is $\"-4\"$ where the coefficients are 2 and -4 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 2 and the last coefficient -4 to get -8. Now what two numbers multiply to -8 and add to the middle coefficient -7? Let's list all of the factors of -8:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -8:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8 ...List the negative factors as well. This will allow us to find all possible combinations\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -8\r
\n" ); document.write( "\n" ); document.write( "(1)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-4)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(8)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(4)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -7\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-8	1+(-8)=-7
2	-4	2+(-4)=-2
-1	8	-1+8=7
-2	4	-2+4=2

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that 1 and -8 add up to -7 and multiply to -8\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"2x%5E2-7x-4\"$ , replace $\"-7x\"$ with $\"1x%2B-8x\"$ (notice $\"1x%2B-8x\"$ adds up to $\"-7x\"$ . So it is equivalent to $\"-7x\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%281x%2B-8x%29%2B-4\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"2x%5E2%2B1x-8x-4\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2%2B1x%29%2B%28-8x-4%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B1%29-4%282x%2B1%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-4\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-4%29%282x%2B1%29\"$ Since we have a common term of $\"2x%2B1\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2%2B1x-8x-4\"$ factors to $\"%28x-4%29%282x%2B1%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"2x%5E2-7x-4\"$ factors to $\"%28x-4%29%282x%2B1%29\"$ (since $\"2x%5E2-7x-4\"$ is equivalent to $\"2x%5E2%2B1x-8x-4\"$ )\r
\n" ); document.write( "
\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( "
\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"4%282x%5E2-7x-4%29\"$ and factors further to $\"4%28x-4%29%282x%2B1%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------\r
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"8x%5E2-16-28x\"$ completely factors to $\"4%28x-4%29%282x%2B1%29\"$ \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );