document.write( "Question 173111: So I have a few problems that i cannot figure out. I need to factor these out. The first one is x^2-15wy+3xy-5wx
\n" ); document.write( "The second one is 9x^2+2x-8
\n" ); document.write( "And the third is 24x^2-10x-4\r
\n" ); document.write( "\n" ); document.write( "I really need help with these. I have seen other examples and still can't figure it out. Could you please help me. Thank you. \n" ); document.write( "

Algebra.Com's Answer #127952 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
# 1\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2-15wy%2B3xy-5wx\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B3xy%29%2B%28-5wx-15wy%29\"$ Rearrange and group like terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3y%29%2B%28-5wx-15wy%29\"$ Factor the GCF \"x\" from the first group\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3y%29-5w%28x%2B3y%29\"$ Factor the GCF -5w from the second group\r
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\n" ); document.write( "\n" ); document.write( "Since we have the common term $\"x%2B3y\"$ we can combine like terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-5w%29%28x%2B3y%29\"$ Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-15wy%2B3xy-5wx\"$ factors to $\"%28x-5w%29%28x%2B3y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2%2B2x-8\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last term $\"-8\"$ to get $\"%289%29%28-8%29=-72\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-72\"$ (the previous product) and add to the second coefficient $\"2\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-72\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-72\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,18,24,36,72\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-72\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-72)
\n" ); document.write( "2*(-36)
\n" ); document.write( "3*(-24)
\n" ); document.write( "4*(-18)
\n" ); document.write( "6*(-12)
\n" ); document.write( "8*(-9)
\n" ); document.write( "(-1)*(72)
\n" ); document.write( "(-2)*(36)
\n" ); document.write( "(-3)*(24)
\n" ); document.write( "(-4)*(18)
\n" ); document.write( "(-6)*(12)
\n" ); document.write( "(-8)*(9)\r
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\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 $\"2\"$ :\r
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First Number	Second Number	Sum
1	-72	1+(-72)=-71
2	-36	2+(-36)=-34
3	-24	3+(-24)=-21
4	-18	4+(-18)=-14
6	-12	6+(-12)=-6
8	-9	8+(-9)=-1
-1	72	-1+72=71
-2	36	-2+36=34
-3	24	-3+24=21
-4	18	-4+18=14
-6	12	-6+12=6
-8	9	-8+9=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that there are no pairs of numbers which add to $\"2\"$ . So $\"9x%5E2%2B2x-8\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "# 3\r
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\n" ); document.write( "\n" ); document.write( " $\"24x%5E2-10x-4\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%2812x%5E2-5x-2%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"12x%5E2-5x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12x%5E2-5x-2\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last term $\"-2\"$ to get $\"%2812%29%28-2%29=-24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-24\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-24\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\r
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\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
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-24)
\n" ); document.write( "2*(-12)
\n" ); document.write( "3*(-8)
\n" ); document.write( "4*(-6)
\n" ); document.write( "(-1)*(24)
\n" ); document.write( "(-2)*(12)
\n" ); document.write( "(-3)*(8)
\n" ); document.write( "(-4)*(6)\r
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\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
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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"-8\"$ add to $\"-5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-8\"$ both multiply to $\"-24\"$ and add to $\"-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5x\"$ with $\"3x-8x\"$ . Remember, $\"3\"$ and $\"-8\"$ add to $\"-5\"$ . So this shows us that $\"3x-8x=-5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2%2Bhighlight%283x-8x%29-2\"$ Replace the second term $\"-5x\"$ with $\"3x-8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812x%5E2%2B3x%29%2B%28-8x-2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2B1%29%2B%28-8x-2%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2B1%29-2%284x%2B1%29\"$ Factor out $\"2\"$ 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
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\n" ); document.write( "\n" ); document.write( " $\"%283x-2%29%284x%2B1%29\"$ Combine like terms. Or factor out the common term $\"4x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E2-5x-2\"$ factors to $\"%283x-2%29%284x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2%2812x%5E2-5x-2%29\"$ and factors further to $\"2%283x-2%29%284x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"24x%5E2-10x-4\"$ factors to $\"2%283x-2%29%284x%2B1%29\"$ \r
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