document.write( "Question 148860: 1)5x^2-10x Factor\r
\n" ); document.write( "\n" ); document.write( "5x(x-5) Is this correct?\r
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\n" ); document.write( "\n" ); document.write( "2)6x^2y^2+12xy^2+12y^2 Factor\r
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\n" ); document.write( "\n" ); document.write( "4) a^2+2a-24 Factor\r
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\n" ); document.write( "\n" ); document.write( "5) 4b^2-28b+49\r
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\n" ); document.write( "\n" ); document.write( "6)3m^3+27m Factor
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Algebra.Com's Answer #109199 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"5x%5E2-10x\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%28x-2%29\"$ Factor out the GCF $\"5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( "2) \r
\n" ); document.write( "\n" ); document.write( "Correct. You can check the answer by distributing\r
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\n" ); document.write( "\n" ); document.write( "Good so far, but can factor $\"a%5E2-b%5E2\"$ into $\"%28a%2Bb%29%28a-b%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3a%5E3b-3ab%5E3\"$ completely factors to $\"3ab%28a%2Bb%29%28a-b%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"a%5E2%2B2a-24\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-24\"$ to get $\"%281%29%28-24%29=-24\"$ .\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 $\"2\"$ ?\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 $\"-24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-24\"$ :\r
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\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( "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 $\"-24\"$ . For instance, $\"1%2A24=-24\"$ , $\"2%2A12=-24\"$ , etc.\r
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\n" ); document.write( "\n" ); document.write( "Since $\"-24\"$ is negative, this means that one factor is positive and one is negative.\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	-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 $\"-4\"$ and $\"6\"$ add to $\"2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"6\"$ both multiply to $\"-24\"$ and add to $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"2a\"$ with $\"-4a%2B6a\"$ . Remember, $\"-4\"$ and $\"6\"$ add to $\"2\"$ . So this shows us that $\"-4a%2B6a=2a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E2%2Bhighlight%28-4a%2B6a%29-24\"$ Replace the second term $\"2a\"$ with $\"-4a%2B6a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2-4a%29%2B%286a-24%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a-4%29%2B%286a-24%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a-4%29%2B6%28a-4%29\"$ Factor out $\"6\"$ 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( " $\"%28a%2B6%29%28a-4%29\"$ Combine like terms. Or factor out the common term $\"a-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"a%5E2%2B2a-24\"$ factors to $\"%28a%2B6%29%28a-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28a%2B6%29%28a-4%29\"$ to get $\"a%5E2%2B2a-24\"$ 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( "Looking at the expression $\"4b%5E2-28b%2B49\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-28\"$ , and the last term is $\"49\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"49\"$ to get $\"%284%29%2849%29=196\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"196\"$ (the previous product) and add to the second coefficient $\"-28\"$ ?\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 $\"196\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"196\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-14,-28,-49,-98,-196\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 $\"196\"$ . For instance, $\"1%2A196=196\"$ , $\"2%2A98=196\"$ , etc.\r
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\n" ); document.write( "\n" ); document.write( "Since $\"196\"$ is positive, this means that either\r
\n" ); document.write( "\n" ); document.write( "a) both factors are positive, or...
\n" ); document.write( "b) both factors are negative.\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 $\"-28\"$ :\r
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First Number	Second Number	Sum
1	196	1+196=197
2	98	2+98=100
4	49	4+49=53
7	28	7+28=35
14	14	14+14=28
-1	-196	-1+(-196)=-197
-2	-98	-2+(-98)=-100
-4	-49	-4+(-49)=-53
-7	-28	-7+(-28)=-35
-14	-14	-14+(-14)=-28

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-14\"$ and $\"-14\"$ add to $\"-28\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-14\"$ and $\"-14\"$ both multiply to $\"196\"$ and add to $\"-28\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-28b\"$ with $\"-14b-14b\"$ . Remember, $\"-14\"$ and $\"-14\"$ add to $\"-28\"$ . So this shows us that $\"-14b-14b=-28b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4b%5E2%2Bhighlight%28-14b-14b%29%2B49\"$ Replace the second term $\"-28b\"$ with $\"-14b-14b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284b%5E2-14b%29%2B%28-14b%2B49%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2b%282b-7%29%2B%28-14b%2B49%29\"$ Factor out the GCF $\"2b\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2b%282b-7%29-7%282b-7%29\"$ Factor out $\"7\"$ 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( " $\"%282b-7%29%282b-7%29\"$ Combine like terms. Or factor out the common term $\"2b-7\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%282b-7%29%5E2\"$ Collect and condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"4b%5E2-28b%2B49\"$ factors to $\"%282b-7%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282b-7%29%5E2\"$ to get $\"4b%5E2-28b%2B49\"$ 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( "6) Correct. You can check the answer by distributing
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