document.write( "Question 253634: Can you help me factor each of the following completely?
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Algebra.Com's Answer #185988 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll do the first two to get you going.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4x%5E2-48x%2B135\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-48\"$ , and the last term is $\"135\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"135\"$ to get $\"%284%29%28135%29=540\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"540\"$ (the previous product) and add to the second coefficient $\"-48\"$ ?\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 $\"540\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"540\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,9,10,12,15,18,20,27,30,36,45,54,60,90,108,135,180,270,540\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-27,-30,-36,-45,-54,-60,-90,-108,-135,-180,-270,-540\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 $\"540\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*540 = 540
\n" ); document.write( "2*270 = 540
\n" ); document.write( "3*180 = 540
\n" ); document.write( "4*135 = 540
\n" ); document.write( "5*108 = 540
\n" ); document.write( "6*90 = 540
\n" ); document.write( "9*60 = 540
\n" ); document.write( "10*54 = 540
\n" ); document.write( "12*45 = 540
\n" ); document.write( "15*36 = 540
\n" ); document.write( "18*30 = 540
\n" ); document.write( "20*27 = 540
\n" ); document.write( "(-1)*(-540) = 540
\n" ); document.write( "(-2)*(-270) = 540
\n" ); document.write( "(-3)*(-180) = 540
\n" ); document.write( "(-4)*(-135) = 540
\n" ); document.write( "(-5)*(-108) = 540
\n" ); document.write( "(-6)*(-90) = 540
\n" ); document.write( "(-9)*(-60) = 540
\n" ); document.write( "(-10)*(-54) = 540
\n" ); document.write( "(-12)*(-45) = 540
\n" ); document.write( "(-15)*(-36) = 540
\n" ); document.write( "(-18)*(-30) = 540
\n" ); document.write( "(-20)*(-27) = 540\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 $\"-48\"$ :\r
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First Number	Second Number	Sum
1	540	1+540=541
2	270	2+270=272
3	180	3+180=183
4	135	4+135=139
5	108	5+108=113
6	90	6+90=96
9	60	9+60=69
10	54	10+54=64
12	45	12+45=57
15	36	15+36=51
18	30	18+30=48
20	27	20+27=47
-1	-540	-1+(-540)=-541
-2	-270	-2+(-270)=-272
-3	-180	-3+(-180)=-183
-4	-135	-4+(-135)=-139
-5	-108	-5+(-108)=-113
-6	-90	-6+(-90)=-96
-9	-60	-9+(-60)=-69
-10	-54	-10+(-54)=-64
-12	-45	-12+(-45)=-57
-15	-36	-15+(-36)=-51
-18	-30	-18+(-30)=-48
-20	-27	-20+(-27)=-47

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-18\"$ and $\"-30\"$ add to $\"-48\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-18\"$ and $\"-30\"$ both multiply to $\"540\"$ and add to $\"-48\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-48x\"$ with $\"-18x-30x\"$ . Remember, $\"-18\"$ and $\"-30\"$ add to $\"-48\"$ . So this shows us that $\"-18x-30x=-48x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%28-18x-30x%29%2B135\"$ Replace the second term $\"-48x\"$ with $\"-18x-30x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2-18x%29%2B%28-30x%2B135%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%282x-9%29%2B%28-30x%2B135%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%282x-9%29-15%282x-9%29\"$ Factor out $\"15\"$ 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( " $\"%282x-15%29%282x-9%29\"$ Combine like terms. Or factor out the common term $\"2x-9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-48x%2B135\"$ factors to $\"%282x-15%29%282x-9%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"4x%5E2-48x%2B135=%282x-15%29%282x-9%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282x-15%29%282x-9%29\"$ to get $\"4x%5E2-48x%2B135\"$ or by graphing the original expression and the answer (the two graphs should be identical).\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 the expression $\"18x%5E2-3x-1\"$ , we can see that the first coefficient is $\"18\"$ , the second coefficient is $\"-3\"$ , and the last term is $\"-1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"18\"$ by the last term $\"-1\"$ to get $\"%2818%29%28-1%29=-18\"$ .\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 $\"-3\"$ ?\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
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-18\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-18) = -18
\n" ); document.write( "2*(-9) = -18
\n" ); document.write( "3*(-6) = -18
\n" ); document.write( "(-1)*(18) = -18
\n" ); document.write( "(-2)*(9) = -18
\n" ); document.write( "(-3)*(6) = -18\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 $\"-3\"$ :\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 the two numbers $\"3\"$ and $\"-6\"$ add to $\"-3\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-6\"$ both multiply to $\"-18\"$ and add to $\"-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-3x\"$ with $\"3x-6x\"$ . Remember, $\"3\"$ and $\"-6\"$ add to $\"-3\"$ . So this shows us that $\"3x-6x=-3x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"18x%5E2%2Bhighlight%283x-6x%29-1\"$ Replace the second term $\"-3x\"$ with $\"3x-6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2818x%5E2%2B3x%29%2B%28-6x-1%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%286x%2B1%29%2B%28-6x-1%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3x%286x%2B1%29-1%286x%2B1%29\"$ Factor out $\"1\"$ 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( "\n" ); document.write( " $\"%283x-1%29%286x%2B1%29\"$ Combine like terms. Or factor out the common term $\"6x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"18x%5E2-3x-1\"$ factors to $\"%283x-1%29%286x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"18x%5E2-3x-1=%283x-1%29%286x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283x-1%29%286x%2B1%29\"$ to get $\"18x%5E2-3x-1\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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