document.write( "Question 173375: Please help Im stuck on these problems. Factoring\r
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Algebra.Com's Answer #128224 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"15a%5E3-30a%5E4-5a%5E5\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"-5a%5E5-30a%5E4%2B15a%5E3\"$ Rearrange the terms in descending order\r
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\n" ); document.write( "\n" ); document.write( " $\"-5a%5E3%28a%5E2%2B6a-3%29\"$ Factor out the GCF $\"-5a%5E3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"a%5E2%2B6a-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"a%5E2%2B6a-3\"$ we can see that the first term is $\"a%5E2\"$ and the last term is $\"-3\"$ where the coefficients are 1 and -3 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -3 to get -3. Now what two numbers multiply to -3 and add to the middle coefficient 6? Let's list all of the factors of -3:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -3:\r
\n" ); document.write( "\n" ); document.write( "1,3\r
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\n" ); document.write( "\n" ); document.write( "-1,-3 ...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 -3\r
\n" ); document.write( "\n" ); document.write( "(1)*(-3)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(3)\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 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6\r
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First Number	Second Number	Sum
1	-3	1+(-3)=-2
-1	3	-1+3=2

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\n" ); document.write( "\n" ); document.write( "None of these pairs of factors add to 6. So the expression $\"1a%5E2%2B6a-3\"$ cannot be factored\r
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\n" ); document.write( "\n" ); document.write( "So $\"-5a%5E5-30a%5E4%2B15a%5E3\"$ factors to $\"-5a%5E3%28a%5E2%2B6a-3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2-8%2B7x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2B7x-8\"$ Rearrange the terms.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B7x-8\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-8\"$ to get $\"%281%29%28-8%29=-8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-8\"$ (the previous product) and add to the second coefficient $\"7\"$ ?\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 $\"-8\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-8\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8\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 $\"-8\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-8)
\n" ); document.write( "2*(-4)
\n" ); document.write( "(-1)*(8)
\n" ); document.write( "(-2)*(4)\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 $\"7\"$ :\r
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First Number	Second Number	Sum
1	-8	1+(-8)=-7
2	-4	2+(-4)=-2
-1	8	-1+8=7
-2	4	-2+4=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"8\"$ add to $\"7\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"8\"$ both multiply to $\"-8\"$ and add to $\"7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"7x\"$ with $\"-x%2B8x\"$ . Remember, $\"-1\"$ and $\"8\"$ add to $\"7\"$ . So this shows us that $\"-x%2B8x=7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-x%2B8x%29-8\"$ Replace the second term $\"7x\"$ with $\"-x%2B8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-x%29%2B%288x-8%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-1%29%2B%288x-8%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-1%29%2B8%28x-1%29\"$ Factor out $\"8\"$ 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( " $\"%28x%2B8%29%28x-1%29\"$ Combine like terms. Or factor out the common term $\"x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B7x-8\"$ factors to $\"%28x%2B8%29%28x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x%2B8%29%28x-1%29\"$ to get $\"x%5E2%2B7x-8\"$ 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( "Since there is nothing in common between the terms of $\"4a-5b%2B6c\"$ and there are no exponents, this means that we cannot factor $\"4a-5b%2B6c\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"49b%5E2-18%2B21b\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"49b%5E2%2B21b-18\"$ Rearrange the terms.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"49b%5E2%2B21b-18\"$ , we can see that the first coefficient is $\"49\"$ , the second coefficient is $\"21\"$ , and the last term is $\"-18\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"49\"$ by the last term $\"-18\"$ to get $\"%2849%29%28-18%29=-882\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-882\"$ (the previous product) and add to the second coefficient $\"21\"$ ?\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 $\"-882\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-882\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,9,14,18,21,42,49,63,98,126,147,294,441,882\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-49,-63,-98,-126,-147,-294,-441,-882\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 $\"-882\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-882)
\n" ); document.write( "2*(-441)
\n" ); document.write( "3*(-294)
\n" ); document.write( "6*(-147)
\n" ); document.write( "7*(-126)
\n" ); document.write( "9*(-98)
\n" ); document.write( "14*(-63)
\n" ); document.write( "18*(-49)
\n" ); document.write( "21*(-42)
\n" ); document.write( "(-1)*(882)
\n" ); document.write( "(-2)*(441)
\n" ); document.write( "(-3)*(294)
\n" ); document.write( "(-6)*(147)
\n" ); document.write( "(-7)*(126)
\n" ); document.write( "(-9)*(98)
\n" ); document.write( "(-14)*(63)
\n" ); document.write( "(-18)*(49)
\n" ); document.write( "(-21)*(42)\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 $\"21\"$ :\r
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First Number	Second Number	Sum
1	-882	1+(-882)=-881
2	-441	2+(-441)=-439
3	-294	3+(-294)=-291
6	-147	6+(-147)=-141
7	-126	7+(-126)=-119
9	-98	9+(-98)=-89
14	-63	14+(-63)=-49
18	-49	18+(-49)=-31
21	-42	21+(-42)=-21
-1	882	-1+882=881
-2	441	-2+441=439
-3	294	-3+294=291
-6	147	-6+147=141
-7	126	-7+126=119
-9	98	-9+98=89
-14	63	-14+63=49
-18	49	-18+49=31
-21	42	-21+42=21

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-21\"$ and $\"42\"$ add to $\"21\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-21\"$ and $\"42\"$ both multiply to $\"-882\"$ and add to $\"21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"21b\"$ with $\"-21b%2B42b\"$ . Remember, $\"-21\"$ and $\"42\"$ add to $\"21\"$ . So this shows us that $\"-21b%2B42b=21b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"49b%5E2%2Bhighlight%28-21b%2B42b%29-18\"$ Replace the second term $\"21b\"$ with $\"-21b%2B42b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2849b%5E2-21b%29%2B%2842b-18%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"7b%287b-3%29%2B%2842b-18%29\"$ Factor out the GCF $\"7b\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"7b%287b-3%29%2B6%287b-3%29\"$ 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( " $\"%287b%2B6%29%287b-3%29\"$ Combine like terms. Or factor out the common term $\"7b-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"49b%5E2%2B21b-18\"$ factors to $\"%287b%2B6%29%287b-3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%287b%2B6%29%287b-3%29\"$ to get $\"49b%5E2%2B21b-18\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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