document.write( "Question 190254: Factor the trinomial\r
\n" ); document.write( "\n" ); document.write( "x^2+18x+45
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Algebra.Com's Answer #142767 by jim_thompson5910(35256)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "Looking at the expression \"x%5E2%2B18x%2B45\", we can see that the first coefficient is \"1\", the second coefficient is \"18\", and the last term is \"45\".\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient \"1\" by the last term \"45\" to get \"%281%29%2845%29=45\".\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to \"45\" (the previous product) and add to the second coefficient \"18\"?\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 \"45\" (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of \"45\":\r
\n" ); document.write( "\n" ); document.write( "1,3,5,9,15,45\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-9,-15,-45\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 \"45\".\r
\n" ); document.write( "\n" ); document.write( "1*45
\n" ); document.write( "3*15
\n" ); document.write( "5*9
\n" ); document.write( "(-1)*(-45)
\n" ); document.write( "(-3)*(-15)
\n" ); document.write( "(-5)*(-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 \"18\":\r
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First NumberSecond NumberSum
1451+45=46
3153+15=18
595+9=14
-1-45-1+(-45)=-46
-3-15-3+(-15)=-18
-5-9-5+(-9)=-14
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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers \"3\" and \"15\" add to \"18\" (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers \"3\" and \"15\" both multiply to \"45\" and add to \"18\"\r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term \"18x\" with \"3x%2B15x\". Remember, \"3\" and \"15\" add to \"18\". So this shows us that \"3x%2B15x=18x\".\r
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\n" ); document.write( "\n" ); document.write( "\"x%5E2%2Bhighlight%283x%2B15x%29%2B45\" Replace the second term \"18x\" with \"3x%2B15x\".\r
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\n" ); document.write( "\n" ); document.write( "\"%28x%5E2%2B3x%29%2B%2815x%2B45%29\" Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( "\"x%28x%2B3%29%2B%2815x%2B45%29\" Factor out the GCF \"x\" from the first group.\r
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\n" ); document.write( "\n" ); document.write( "\"x%28x%2B3%29%2B15%28x%2B3%29\" Factor out \"15\" 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%2B15%29%28x%2B3%29\" Combine like terms. Or factor out the common term \"x%2B3\"\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%2B18x%2B45\" factors to \"%28x%2B15%29%28x%2B3%29\".\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing \"%28x%2B15%29%28x%2B3%29\" to get \"x%5E2%2B18x%2B45\" or by graphing the original expression and the answer (the two graphs should be identical).
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