document.write( "Question 131011: 3a^2+12ab+12b^2 can you show me how this is done completely
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Algebra.Com's Answer #95633 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( " $\"3a%5E2%2B12ab%2B12b%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"3%28a%5E2%2B4ab%2B4b%5E2%29\"$ Factor out the GCF $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"a%5E2%2B4ab%2B4b%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"1a%5E2%2B4ab%2B4b%5E2\"$ we can see that the first term is $\"1a%5E2\"$ and the last term is $\"4b%5E2\"$ where the coefficients are 1 and 4 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 4 to get 4. Now what two numbers multiply to 4 and add to the middle coefficient 4? Let's list all of the factors of 4:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 4:\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 4\r
\n" ); document.write( "\n" ); document.write( "1*4\r
\n" ); document.write( "\n" ); document.write( "2*2\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-4)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-2)\r
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\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( "Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4\r
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First Number	Second Number	Sum
1	4	1+4=5
2	2	2+2=4
-1	-4	-1+(-4)=-5
-2	-2	-2+(-2)=-4

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 2 and 2 add up to 4 and multiply to 4\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"1a%5E2%2B4ab%2B4b%5E2\"$ , replace $\"4ab\"$ with $\"2ab%2B2ab\"$ (notice $\"2ab%2B2ab\"$ adds up to $\"4ab\"$ . So it is equivalent to $\"4ab\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"1a%5E2%2Bhighlight%282ab%2B2ab%29%2B4b%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1a%5E2%2B2ab%2B2ab%2B4b%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%281a%5E2%2B2ab%29%2B%282ab%2B4b%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a%2B2b%29%2B2b%28a%2B2b%29\"$ Factor out the GCF of $\"a\"$ out of the first group. Factor out the GCF of $\"2b\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%2B2b%29%28a%2B2b%29\"$ Since we have a common term of $\"a%2B2b\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"1a%5E2%2B2ab%2B2ab%2B4b%5E2\"$ factors to $\"%28a%2B2b%29%28a%2B2b%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"1a%5E2%2B4ab%2B4b%5E2\"$ factors to $\"%28a%2B2b%29%28a%2B2b%29\"$ (since $\"1a%5E2%2B4ab%2B4b%5E2\"$ is equivalent to $\"1a%5E2%2B2ab%2B2ab%2B4b%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%28a%2B2b%29%28a%2B2b%29\"$ is equivalent to $\"%28a%2B2b%29%5E2\"$ since the term $\"a%2B2b\"$ occurs twice. So $\"1a%5E2%2B4ab%2B4b%5E2\"$ also factors to $\"%28a%2B2b%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"3%28a%5E2%2B4ab%2B4b%5E2%29\"$ and factors further to $\"3%28a%2B2b%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3a%5E2%2B12ab%2B12b%5E2\"$ factors to $\"3%28a%2B2b%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"8x%5E2-26x-15\"$ we can see that the first term is $\"8x%5E2\"$ and the last term is $\"-15\"$ where the coefficients are 8 and -15 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 8 and the last coefficient -15 to get -120. Now what two numbers multiply to -120 and add to the middle coefficient -26? Let's list all of the factors of -120:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -120:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120 ...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 -120\r
\n" ); document.write( "\n" ); document.write( "(1)*(-120)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-40)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(8)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(120)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(60)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(40)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(20)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(15)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(12)\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 -26? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -26\r
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First Number	Second Number	Sum
1	-120	1+(-120)=-119
2	-60	2+(-60)=-58
3	-40	3+(-40)=-37
4	-30	4+(-30)=-26
5	-24	5+(-24)=-19
6	-20	6+(-20)=-14
8	-15	8+(-15)=-7
10	-12	10+(-12)=-2
-1	120	-1+120=119
-2	60	-2+60=58
-3	40	-3+40=37
-4	30	-4+30=26
-5	24	-5+24=19
-6	20	-6+20=14
-8	15	-8+15=7
-10	12	-10+12=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 4 and -30 add up to -26 and multiply to -120\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"8x%5E2-26x-15\"$ , replace $\"-26x\"$ with $\"4x%2B-30x\"$ (notice $\"4x%2B-30x\"$ adds up to $\"-26x\"$ . So it is equivalent to $\"-26x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"8x%5E2%2Bhighlight%284x%2B-30x%29%2B-15\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"8x%5E2%2B4x-30x-15\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%288x%5E2%2B4x%29%2B%28-30x-15%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%282x%2B1%29-15%282x%2B1%29\"$ Factor out the GCF of $\"4x\"$ out of the first group. Factor out the GCF of $\"-15\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x-15%29%282x%2B1%29\"$ Since we have a common term of $\"2x%2B1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"8x%5E2%2B4x-30x-15\"$ factors to $\"%284x-15%29%282x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"8x%5E2-26x-15\"$ factors to $\"%284x-15%29%282x%2B1%29\"$ (since $\"8x%5E2-26x-15\"$ is equivalent to $\"8x%5E2%2B4x-30x-15\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"8x%5E2-26x-15\"$ factors to $\"%284x-15%29%282x%2B1%29\"$
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