document.write( "Question 171506: factor the trinomial x^2+x-12, 9x^2-24x+6, 2x^2+9x-x-35, 4-5x-6x^2 \n" ); document.write( "

Algebra.Com's Answer #126650 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll do the first two to get you started\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2Bx-12\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"1\"$ , and the last term is $\"-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-12\"$ to get $\"%281%29%28-12%29=-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-12\"$ (the previous product) and add to the second coefficient $\"1\"$ ?\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 $\"-12\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-12\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\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 $\"-12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-12)
\n" ); document.write( "2*(-6)
\n" ); document.write( "3*(-4)
\n" ); document.write( "(-1)*(12)
\n" ); document.write( "(-2)*(6)
\n" ); document.write( "(-3)*(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 $\"1\"$ :\r
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First Number	Second Number	Sum
1	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"4\"$ add to $\"1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"4\"$ both multiply to $\"-12\"$ and add to $\"1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"1x\"$ with $\"-3x%2B4x\"$ . Remember, $\"-3\"$ and $\"4\"$ add to $\"1\"$ . So this shows us that $\"-3x%2B4x=1x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-3x%2B4x%29-12\"$ Replace the second term $\"1x\"$ with $\"-3x%2B4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-3x%29%2B%284x-12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-3%29%2B%284x-12%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-3%29%2B4%28x-3%29\"$ Factor out $\"4\"$ 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( " $\"%28x%2B4%29%28x-3%29\"$ Combine like terms. Or factor out the common term $\"x-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2Bx-12\"$ factors to $\"%28x%2B4%29%28x-3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x%2B4%29%28x-3%29\"$ to get $\"x%5E2%2Bx-12\"$ 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( "# 2\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2-24x%2B6\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"-24\"$ , and the last term is $\"6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last term $\"6\"$ to get $\"%289%29%286%29=54\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"54\"$ (the previous product) and add to the second coefficient $\"-24\"$ ?\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 $\"54\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"54\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,9,18,27,54\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-9,-18,-27,-54\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 $\"54\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*54
\n" ); document.write( "2*27
\n" ); document.write( "3*18
\n" ); document.write( "6*9
\n" ); document.write( "(-1)*(-54)
\n" ); document.write( "(-2)*(-27)
\n" ); document.write( "(-3)*(-18)
\n" ); document.write( "(-6)*(-9)\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 $\"-24\"$ :\r
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First Number	Second Number	Sum
1	54	1+54=55
2	27	2+27=29
3	18	3+18=21
6	9	6+9=15
-1	-54	-1+(-54)=-55
-2	-27	-2+(-27)=-29
-3	-18	-3+(-18)=-21
-6	-9	-6+(-9)=-15

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that there are no pairs of numbers which add to $\"-24\"$ . \r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2-24x%2B6\"$ cannot be factored. This means that the polynomial is prime.
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