document.write( "Question 153303: Factor the expression 4n^3 + 8n^2 - 5n - 10\r
\n" ); document.write( "\n" ); document.write( "Factor k^2 + kf - 2f^2\r
\n" ); document.write( "\n" ); document.write( "Factor 6g^2 + 11g - 35\r
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\n" ); document.write( "\n" ); document.write( "I am not sure how to factor these. I am especially \"stuck\" on the one that uses kf.\r
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Algebra.Com's Answer #112828 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( " $\"4n%5E3%2B8n%5E2-5n-10\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"%284n%5E3%2B8n%5E2%29%2B%28-5n-10%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"4n%5E2%28n%2B2%29-5%28n%2B2%29\"$ Factor out the GCF $\"4n%5E2\"$ out of the first group. Factor out the GCF $\"-5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%284n%5E2-5%29%28n%2B2%29\"$ Since we have the common term $\"n%2B2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"4n%5E3%2B8n%5E2-5n-10\"$ factors to $\"%284n%5E2-5%29%28n%2B2%29\"$ \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 $\"k%5E2%2Bkf-2f%5E2\"$ we can see that the first term is $\"k%5E2\"$ and the last term is $\"-2f%5E2\"$ where the coefficients are 1 and -2 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -2 to get -2. Now what two numbers multiply to -2 and add to the middle coefficient 1? Let's list all of the factors of -2:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -2:\r
\n" ); document.write( "\n" ); document.write( "1,2\r
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\n" ); document.write( "\n" ); document.write( "-1,-2 ...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 -2\r
\n" ); document.write( "\n" ); document.write( "(1)*(-2)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(2)\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 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1\r
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First Number	Second Number	Sum
1	-2	1+(-2)=-1
-1	2	-1+2=1

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -1 and 2 add up to 1 and multiply to -2\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"k%5E2%2Bkf-2f%5E2\"$ , replace $\"kf\"$ with $\"-kf%2B2kf\"$ (notice $\"-kf%2B2kf\"$ adds up to $\"kf\"$ . So it is equivalent to $\"kf\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"k%5E2%2Bhighlight%28-kf%2B2kf%29%2B-2f%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"k%5E2-kf%2B2kf-2f%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28k%5E2-kf%29%2B%282kf-2f%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"k%28k-f%29%2B2f%28k-f%29\"$ Factor out the GCF of $\"k\"$ out of the first group. Factor out the GCF of $\"2f\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28k%2B2f%29%28k-f%29\"$ Since we have a common term of $\"k-f\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"k%5E2-kf%2B2kf-2f%5E2\"$ factors to $\"%28k%2B2f%29%28k-f%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"k%5E2%2Bkf-2f%5E2\"$ factors to $\"%28k%2B2f%29%28k-f%29\"$ (since $\"k%5E2%2Bkf-2f%5E2\"$ is equivalent to $\"k%5E2-kf%2B2kf-2f%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"k%5E2%2Bkf-2f%5E2\"$ factors to $\"%28k%2B2f%29%28k-f%29\"$ \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 $\"6g%5E2%2B11g-35\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"11\"$ , and the last term is $\"-35\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-35\"$ to get $\"%286%29%28-35%29=-210\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-210\"$ (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 $\"-210\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-210\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210\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 $\"-210\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-210)
\n" ); document.write( "2*(-105)
\n" ); document.write( "3*(-70)
\n" ); document.write( "5*(-42)
\n" ); document.write( "6*(-35)
\n" ); document.write( "7*(-30)
\n" ); document.write( "10*(-21)
\n" ); document.write( "14*(-15)
\n" ); document.write( "(-1)*(210)
\n" ); document.write( "(-2)*(105)
\n" ); document.write( "(-3)*(70)
\n" ); document.write( "(-5)*(42)
\n" ); document.write( "(-6)*(35)
\n" ); document.write( "(-7)*(30)
\n" ); document.write( "(-10)*(21)
\n" ); document.write( "(-14)*(15)\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	-210	1+(-210)=-209
2	-105	2+(-105)=-103
3	-70	3+(-70)=-67
5	-42	5+(-42)=-37
6	-35	6+(-35)=-29
7	-30	7+(-30)=-23
10	-21	10+(-21)=-11
14	-15	14+(-15)=-1
-1	210	-1+210=209
-2	105	-2+105=103
-3	70	-3+70=67
-5	42	-5+42=37
-6	35	-6+35=29
-7	30	-7+30=23
-10	21	-10+21=11
-14	15	-14+15=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-10\"$ and $\"21\"$ add to $\"11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-10\"$ and $\"21\"$ both multiply to $\"-210\"$ and add to $\"11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"11g\"$ with $\"-10g%2B21g\"$ . Remember, $\"-10\"$ and $\"21\"$ add to $\"11\"$ . So this shows us that $\"-10g%2B21g=11g\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6g%5E2%2Bhighlight%28-10g%2B21g%29-35\"$ Replace the second term $\"11g\"$ with $\"-10g%2B21g\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286g%5E2-10g%29%2B%2821g-35%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2g%283g-5%29%2B%2821g-35%29\"$ Factor out the GCF $\"2g\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2g%283g-5%29%2B7%283g-5%29\"$ 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( " $\"%282g%2B7%29%283g-5%29\"$ Combine like terms. Or factor out the common term $\"3g-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6g%5E2%2B11g-35\"$ factors to $\"%282g%2B7%29%283g-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282g%2B7%29%283g-5%29\"$ to get $\"6g%5E2%2B11g-35\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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