document.write( "Question 173367: please help with factoring
\n" ); document.write( "3x^2y^3-3x^2y\r
\n" ); document.write( "\n" ); document.write( "x^2+144\r
\n" ); document.write( "\n" ); document.write( "5x^2-14x+16\r
\n" ); document.write( "\n" ); document.write( "2x^2+11x-9
\n" ); document.write( "Thank you. \n" ); document.write( "

Algebra.Com's Answer #128218 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( " $\"3x%5E2y%5E3-3x%5E2y\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2y%28y%5E2-1%29\"$ Factor out the GCF $\"3x%5E2y\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2y%28y%2B1%29%28y-1%29\"$ Factor $\"y%5E2-1\"$ to get $\"%28y%2B1%29%28y-1%29\"$ (by using the difference of squares)\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2y%5E3-3x%5E2y\"$ factors to $\"3x%5E2y%28y%2B1%29%28y-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Note: the expression $\"x%5E2%2B144\"$ really looks like $\"x%5E2%2B0x%2B144\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2%2B0x%2B144\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"144\"$ where the coefficients are 1 and 144 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 144 to get 144. Now what two numbers multiply to 144 and add to the middle coefficient 0? Let's list all of the factors of 144:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 144:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,36,48,72\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. 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 144\r
\n" ); document.write( "\n" ); document.write( "1*144\r
\n" ); document.write( "\n" ); document.write( "2*72\r
\n" ); document.write( "\n" ); document.write( "3*48\r
\n" ); document.write( "\n" ); document.write( "4*36\r
\n" ); document.write( "\n" ); document.write( "6*24\r
\n" ); document.write( "\n" ); document.write( "8*18\r
\n" ); document.write( "\n" ); document.write( "9*16\r
\n" ); document.write( "\n" ); document.write( "12*12\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-144)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-72)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(-12)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 0? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 0\r
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First Number	Second Number	Sum
1	144	1+144=145
2	72	2+72=74
3	48	3+48=51
4	36	4+36=40
6	24	6+24=30
8	18	8+18=26
9	16	9+16=25
12	12	12+12=24
-1	-144	-1+(-144)=-145
-2	-72	-2+(-72)=-74
-3	-48	-3+(-48)=-51
-4	-36	-4+(-36)=-40
-6	-24	-6+(-24)=-30
-8	-18	-8+(-18)=-26
-9	-16	-9+(-16)=-25
-12	-12	-12+(-12)=-24

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\n" ); document.write( "\n" ); document.write( "None of these pairs of factors add to 0. So the expression $\"x%5E2%2B0x%2B144\"$ 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( "Looking at the expression $\"5x%5E2-14x%2B16\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"-14\"$ , and the last term is $\"16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"16\"$ to get $\"%285%29%2816%29=80\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"80\"$ (the previous product) and add to the second coefficient $\"-14\"$ ?\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 $\"80\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"80\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,16,20,40,80\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-16,-20,-40,-80\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 $\"80\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*80
\n" ); document.write( "2*40
\n" ); document.write( "4*20
\n" ); document.write( "5*16
\n" ); document.write( "8*10
\n" ); document.write( "(-1)*(-80)
\n" ); document.write( "(-2)*(-40)
\n" ); document.write( "(-4)*(-20)
\n" ); document.write( "(-5)*(-16)
\n" ); document.write( "(-8)*(-10)\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 $\"-14\"$ :\r
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First Number	Second Number	Sum
1	80	1+80=81
2	40	2+40=42
4	20	4+20=24
5	16	5+16=21
8	10	8+10=18
-1	-80	-1+(-80)=-81
-2	-40	-2+(-40)=-42
-4	-20	-4+(-20)=-24
-5	-16	-5+(-16)=-21
-8	-10	-8+(-10)=-18

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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 $\"-14\"$ . \r
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\n" ); document.write( "\n" ); document.write( "So $\"5x%5E2-14x%2B16\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "# 4\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2%2B11x-9\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"11\"$ , and the last term is $\"-9\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"-9\"$ to get $\"%282%29%28-9%29=-18\"$ .\r
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\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 $\"11\"$ ?\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 $\"-18\"$ (the previous product).\r
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\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
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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 $\"-18\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-18)
\n" ); document.write( "2*(-9)
\n" ); document.write( "3*(-6)
\n" ); document.write( "(-1)*(18)
\n" ); document.write( "(-2)*(9)
\n" ); document.write( "(-3)*(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 $\"11\"$ :\r
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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

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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 $\"11\"$ . So $\"2x%5E2%2B11x-9\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "-------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Note: there's always the possibility that you cannot factor an expression. However, most books will only throw them out in small doses. So I would double check your problems to make sure that you copied them down correctly. \n" ); document.write( "

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