document.write( "Question 115655: Can someone please help me with this problem. I tried to do the problem and I just keep getting very confused.\r
\n" ); document.write( "\n" ); document.write( "We are specifically looking for how to factor polynomials when the leading coefficient is not 1. How can we find the correct way to break up the middle term to make this come out right.\r
\n" ); document.write( "\n" ); document.write( "Can you please factor the polynomial\r
\n" ); document.write( "\n" ); document.write( "4x^2 + 4x - 99\r
\n" ); document.write( "\n" ); document.write( "Please tell us why and how you carry out each step of the process.\r
\n" ); document.write( "\n" ); document.write( "PLEASE HELP ME!!!!
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #84166 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at $\"4x%5E2%2B4x-99\"$ we can see that the first term is $\"4x%5E2\"$ and the last term is $\"-99\"$ where the coefficients are 4 and -99 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 4 and the last coefficient -99 to get -396. Now what two numbers multiply to -396 and add to the middle coefficient 4? Let's list all of the factors of -396:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -396:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,11,12,18,22,33,36,44,66,99,132,198,396\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-11,-12,-18,-22,-33,-36,-44,-66,-99,-132,-198,-396 ...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 -396\r
\n" ); document.write( "\n" ); document.write( "(1)*(-396)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-198)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-132)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-99)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-66)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-44)\r
\n" ); document.write( "\n" ); document.write( "(11)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(12)*(-33)\r
\n" ); document.write( "\n" ); document.write( "(18)*(-22)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(396)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(198)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(132)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(99)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(66)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(44)\r
\n" ); document.write( "\n" ); document.write( "(-11)*(36)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(33)\r
\n" ); document.write( "\n" ); document.write( "(-18)*(22)\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 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-396	1+(-396)=-395
2	-198	2+(-198)=-196
3	-132	3+(-132)=-129
4	-99	4+(-99)=-95
6	-66	6+(-66)=-60
9	-44	9+(-44)=-35
11	-36	11+(-36)=-25
12	-33	12+(-33)=-21
18	-22	18+(-22)=-4
-1	396	-1+396=395
-2	198	-2+198=196
-3	132	-3+132=129
-4	99	-4+99=95
-6	66	-6+66=60
-9	44	-9+44=35
-11	36	-11+36=25
-12	33	-12+33=21
-18	22	-18+22=4

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that -18 and 22 add up to 4 and multiply to -396\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"4x%5E2%2B4x-99\"$ , replace $\"4x\"$ with $\"-18x%2B22x\"$ (notice $\"-18x%2B22x\"$ adds up to $\"4x\"$ . So it is equivalent to $\"4x\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%28-18x%2B22x%29%2B-99\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"4x%5E2-18x%2B22x-99\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2-18x%29%2B%2822x-99%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%282x-9%29%2B11%282x-9%29\"$ Factor out the GCF of $\"2x\"$ out of the first group. Factor out the GCF of $\"11\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%282x%2B11%29%282x-9%29\"$ Since we have a common term of $\"2x-9\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-18x%2B22x-99\"$ factors to $\"%282x%2B11%29%282x-9%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"4x%5E2%2B4x-99\"$ factors to $\"%282x%2B11%29%282x-9%29\"$ (since $\"4x%5E2%2B4x-99\"$ is equivalent to $\"4x%5E2-18x%2B22x-99\"$ )\r
\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 $\"4x%5E2%2B4x-99\"$ factors to $\"%282x%2B11%29%282x-9%29\"$ \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );