document.write( "Question 198862: Factor the algebraic expression completely x^2-16x+63 \n" ); document.write( "

Algebra.Com's Answer #149349 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-16x%2B63\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-16\"$ , and the last term is $\"63\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"63\"$ to get $\"%281%29%2863%29=63\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"63\"$ (the previous product) and add to the second coefficient $\"-16\"$ ?\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 $\"63\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"63\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,7,9,21,63\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-9,-21,-63\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 $\"63\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*63
\n" ); document.write( "3*21
\n" ); document.write( "7*9
\n" ); document.write( "(-1)*(-63)
\n" ); document.write( "(-3)*(-21)
\n" ); document.write( "(-7)*(-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 $\"-16\"$ :\r
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First Number	Second Number	Sum
1	63	1+63=64
3	21	3+21=24
7	9	7+9=16
-1	-63	-1+(-63)=-64
-3	-21	-3+(-21)=-24
-7	-9	-7+(-9)=-16

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-7\"$ and $\"-9\"$ add to $\"-16\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-7\"$ and $\"-9\"$ both multiply to $\"63\"$ and add to $\"-16\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-16x\"$ with $\"-7x-9x\"$ . Remember, $\"-7\"$ and $\"-9\"$ add to $\"-16\"$ . So this shows us that $\"-7x-9x=-16x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-7x-9x%29%2B63\"$ Replace the second term $\"-16x\"$ with $\"-7x-9x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-7x%29%2B%28-9x%2B63%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-7%29%2B%28-9x%2B63%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-7%29-9%28x-7%29\"$ Factor out $\"9\"$ 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( " $\"%28x-9%29%28x-7%29\"$ Combine like terms. Or factor out the common term $\"x-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-16x%2B63\"$ factors to $\"%28x-9%29%28x-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x-9%29%28x-7%29\"$ to get $\"x%5E2-16x%2B63\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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