document.write( "Question 190254: Factor the trinomial\r
\n" ); document.write( "\n" ); document.write( "x^2+18x+45 \n" ); document.write( "

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 Number	Second Number	Sum
1	45	1+45=46
3	15	3+15=18
5	9	5+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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