document.write( "Question 191799: factor completely\r
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\n" ); document.write( "\n" ); document.write( "x^2+22x+57 \n" ); document.write( "

Algebra.Com's Answer #143906 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B22x%2B57\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"22\"$ , and the last term is $\"57\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"57\"$ to get $\"%281%29%2857%29=57\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"57\"$ (the previous product) and add to the second coefficient $\"22\"$ ?\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 $\"57\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"57\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-19,-57\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 $\"57\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*57
\n" ); document.write( "3*19
\n" ); document.write( "(-1)*(-57)
\n" ); document.write( "(-3)*(-19)\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 $\"22\"$ :\r
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First Number	Second Number	Sum
1	57	1+57=58
3	19	3+19=22
-1	-57	-1+(-57)=-58
-3	-19	-3+(-19)=-22

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"19\"$ add to $\"22\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"19\"$ both multiply to $\"57\"$ and add to $\"22\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"22x\"$ with $\"3x%2B19x\"$ . Remember, $\"3\"$ and $\"19\"$ add to $\"22\"$ . So this shows us that $\"3x%2B19x=22x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%283x%2B19x%29%2B57\"$ Replace the second term $\"22x\"$ with $\"3x%2B19x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B3x%29%2B%2819x%2B57%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B%2819x%2B57%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B19%28x%2B3%29\"$ Factor out $\"19\"$ 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( " $\"%28x%2B19%29%28x%2B3%29\"$ 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 $\"x%5E2%2B22x%2B57\"$ factors to $\"%28x%2B19%29%28x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x%2B19%29%28x%2B3%29\"$ to get $\"x%5E2%2B22x%2B57\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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