document.write( "Question 114003: Factoring these problems ? .\r
\n" ); document.write( "\n" ); document.write( "1- 18xy^3 + 3xy^2 - 10xy.\r
\n" ); document.write( "\n" ); document.write( "2- 15x^2 + 7x - 2.\r
\n" ); document.write( "\n" ); document.write( "3- 25x^2 + 20x + 4. \n" ); document.write( "

Algebra.Com's Answer #82967 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( " $\"18xy%5E3%2B3xy%5E2-10xy\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"xy%2818y%5E2%2B3y-10%29\"$ Factor out the GCF $\"xy\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"18y%5E2%2B3y-10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"18y%5E2%2B3y-10\"$ we can see that the first term is $\"18y%5E2\"$ and the last term is $\"-10\"$ where the coefficients are 18 and -10 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 18 and the last coefficient -10 to get -180. Now what two numbers multiply to -180 and add to the middle coefficient 3? Let's list all of the factors of -180:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -180:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-30,-36,-45,-60,-90,-180 ...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 -180\r
\n" ); document.write( "\n" ); document.write( "(1)*(-180)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-90)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-45)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(12)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(180)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(90)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(60)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(45)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(36)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(20)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(18)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(15)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3\r
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First Number	Second Number	Sum
1	-180	1+(-180)=-179
2	-90	2+(-90)=-88
3	-60	3+(-60)=-57
4	-45	4+(-45)=-41
5	-36	5+(-36)=-31
6	-30	6+(-30)=-24
9	-20	9+(-20)=-11
10	-18	10+(-18)=-8
12	-15	12+(-15)=-3
-1	180	-1+180=179
-2	90	-2+90=88
-3	60	-3+60=57
-4	45	-4+45=41
-5	36	-5+36=31
-6	30	-6+30=24
-9	20	-9+20=11
-10	18	-10+18=8
-12	15	-12+15=3

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -12 and 15 add up to 3 and multiply to -180\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"18y%5E2%2B3y-10\"$ , replace $\"3y\"$ with $\"-12y%2B15y\"$ (notice $\"-12y%2B15y\"$ adds up to $\"3y\"$ . So it is equivalent to $\"3y\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"18y%5E2%2Bhighlight%28-12y%2B15y%29%2B-10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"18y%5E2-12y%2B15y-10\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2818y%5E2-12y%29%2B%2815y-10%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"6y%283y-2%29%2B5%283y-2%29\"$ Factor out the GCF of $\"6y\"$ out of the first group. Factor out the GCF of $\"5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%286y%2B5%29%283y-2%29\"$ Since we have a common term of $\"3y-2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"18y%5E2%2B3y-10\"$ factors to $\"%286y%2B5%29%283y-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"xy%286y%2B5%29%283y-2%29\"$ Now reintroduce the GCF\r
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\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"18xy%5E3%2B3xy%5E2-10xy\"$ factors to $\"xy%286y%2B5%29%283y-2%29\"$ \r
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\r
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\n" ); document.write( "\n" ); document.write( "#2\r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"15x%5E2%2B7x-2\"$ we can see that the first term is $\"15x%5E2\"$ and the last term is $\"-2\"$ where the coefficients are 15 and -2 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 15 and the last coefficient -2 to get -30. Now what two numbers multiply to -30 and add to the middle coefficient 7? Let's list all of the factors of -30:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -30:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,30\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30 ...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 -30\r
\n" ); document.write( "\n" ); document.write( "(1)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(15)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(10)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(6)\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
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\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
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First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -3 and 10 add up to 7 and multiply to -30\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"15x%5E2%2B7x-2\"$ , replace $\"7x\"$ with $\"-3x%2B10x\"$ (notice $\"-3x%2B10x\"$ adds up to $\"7x\"$ . So it is equivalent to $\"7x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"15x%5E2%2Bhighlight%28-3x%2B10x%29%2B-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"15x%5E2-3x%2B10x-2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2815x%5E2-3x%29%2B%2810x-2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%285x-1%29%2B2%285x-1%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"2\"$ out of the second group\r
\n" ); document.write( "
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B2%29%285x-1%29\"$ Since we have a common term of $\"5x-1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"15x%5E2-3x%2B10x-2\"$ factors to $\"%283x%2B2%29%285x-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"15x%5E2%2B7x-2\"$ factors to $\"%283x%2B2%29%285x-1%29\"$ (since $\"15x%5E2%2B7x-2\"$ is equivalent to $\"15x%5E2-3x%2B10x-2\"$ )\r
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\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 $\"25x%5E2%2B20x%2B4\"$ we can see that the first term is $\"25x%5E2\"$ and the last term is $\"4\"$ where the coefficients are 25 and 4 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 25 and the last coefficient 4 to get 100. Now what two numbers multiply to 100 and add to the middle coefficient 20? Let's list all of the factors of 100:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 100:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,10,20,25,50\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20,-25,-50 ...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 100\r
\n" ); document.write( "\n" ); document.write( "1*100\r
\n" ); document.write( "\n" ); document.write( "2*50\r
\n" ); document.write( "\n" ); document.write( "4*25\r
\n" ); document.write( "\n" ); document.write( "5*20\r
\n" ); document.write( "\n" ); document.write( "10*10\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-100)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-50)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-25)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(-10)\r
\n" ); document.write( "
\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( "\n" ); document.write( "Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20\r
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First Number	Second Number	Sum
1	100	1+100=101
2	50	2+50=52
4	25	4+25=29
5	20	5+20=25
10	10	10+10=20
-1	-100	-1+(-100)=-101
-2	-50	-2+(-50)=-52
-4	-25	-4+(-25)=-29
-5	-20	-5+(-20)=-25
-10	-10	-10+(-10)=-20

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 10 and 10 add up to 20 and multiply to 100\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"25x%5E2%2B20x%2B4\"$ , replace $\"20x\"$ with $\"10x%2B10x\"$ (notice $\"10x%2B10x\"$ adds up to $\"20x\"$ . So it is equivalent to $\"20x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"25x%5E2%2Bhighlight%2810x%2B10x%29%2B4\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"25x%5E2%2B10x%2B10x%2B4\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2825x%5E2%2B10x%29%2B%2810x%2B4%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"5x%285x%2B2%29%2B2%285x%2B2%29\"$ Factor out the GCF of $\"5x\"$ out of the first group. Factor out the GCF of $\"2\"$ out of the second group\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%285x%2B2%29%285x%2B2%29\"$ Since we have a common term of $\"5x%2B2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"25x%5E2%2B10x%2B10x%2B4\"$ factors to $\"%285x%2B2%29%285x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"25x%5E2%2B20x%2B4\"$ factors to $\"%285x%2B2%29%285x%2B2%29\"$ (since $\"25x%5E2%2B20x%2B4\"$ is equivalent to $\"25x%5E2%2B10x%2B10x%2B4\"$ )\r
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