document.write( "Question 476851: factor the algebraic expressions completely\r
\n" ); document.write( "\n" ); document.write( "12x^2 -34x + 24 \n" ); document.write( "

Algebra.Com's Answer #326936 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2-34x%2B24\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%286x%5E2-17x%2B12%29\"$ Factor out the GCF $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"6x%5E2-17x%2B12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2-17x%2B12\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-17\"$ , and the last term is $\"12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"12\"$ to get $\"%286%29%2812%29=72\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"72\"$ (the previous product) and add to the second coefficient $\"-17\"$ ?\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 $\"72\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"72\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,18,24,36,72\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72\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 $\"72\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*72 = 72
\n" ); document.write( "2*36 = 72
\n" ); document.write( "3*24 = 72
\n" ); document.write( "4*18 = 72
\n" ); document.write( "6*12 = 72
\n" ); document.write( "8*9 = 72
\n" ); document.write( "(-1)*(-72) = 72
\n" ); document.write( "(-2)*(-36) = 72
\n" ); document.write( "(-3)*(-24) = 72
\n" ); document.write( "(-4)*(-18) = 72
\n" ); document.write( "(-6)*(-12) = 72
\n" ); document.write( "(-8)*(-9) = 72\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 $\"-17\"$ :\r
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First Number	Second Number	Sum
1	72	1+72=73
2	36	2+36=38
3	24	3+24=27
4	18	4+18=22
6	12	6+12=18
8	9	8+9=17
-1	-72	-1+(-72)=-73
-2	-36	-2+(-36)=-38
-3	-24	-3+(-24)=-27
-4	-18	-4+(-18)=-22
-6	-12	-6+(-12)=-18
-8	-9	-8+(-9)=-17

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-8\"$ and $\"-9\"$ add to $\"-17\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-8\"$ and $\"-9\"$ both multiply to $\"72\"$ and add to $\"-17\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-17x\"$ with $\"-8x-9x\"$ . Remember, $\"-8\"$ and $\"-9\"$ add to $\"-17\"$ . So this shows us that $\"-8x-9x=-17x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%28-8x-9x%29%2B12\"$ Replace the second term $\"-17x\"$ with $\"-8x-9x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2-8x%29%2B%28-9x%2B12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%283x-4%29%2B%28-9x%2B12%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%283x-4%29-3%283x-4%29\"$ Factor out $\"3\"$ 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( " $\"%282x-3%29%283x-4%29\"$ Combine like terms. Or factor out the common term $\"3x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"2%286x%5E2-17x%2B12%29\"$ then factors further to $\"2%282x-3%29%283x-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E2-34x%2B24\"$ completely factors to $\"2%282x-3%29%283x-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"12x%5E2-34x%2B24=2%282x-3%29%283x-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%282x-3%29%283x-4%29\"$ to get $\"12x%5E2-34x%2B24\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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