document.write( "Question 887725: Completely factor the expression.
\n" ); document.write( "x2+10x+16 \n" ); document.write( "

Algebra.Com's Answer #536821 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B10x%2B16\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"10\"$ , and the last term is $\"16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"16\"$ to get $\"%281%29%2816%29=16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"16\"$ (the previous product) and add to the second coefficient $\"10\"$ ?\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 $\"16\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"16\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8,-16\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 $\"16\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*16 = 16
\n" ); document.write( "2*8 = 16
\n" ); document.write( "4*4 = 16
\n" ); document.write( "(-1)*(-16) = 16
\n" ); document.write( "(-2)*(-8) = 16
\n" ); document.write( "(-4)*(-4) = 16\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 $\"10\"$ :\r
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First Number	Second Number	Sum
1	16	1+16=17
2	8	2+8=10
4	4	4+4=8
-1	-16	-1+(-16)=-17
-2	-8	-2+(-8)=-10
-4	-4	-4+(-4)=-8

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"8\"$ add to $\"10\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"8\"$ both multiply to $\"16\"$ and add to $\"10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"10x\"$ with $\"2x%2B8x\"$ . Remember, $\"2\"$ and $\"8\"$ add to $\"10\"$ . So this shows us that $\"2x%2B8x=10x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%282x%2B8x%29%2B16\"$ Replace the second term $\"10x\"$ with $\"2x%2B8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B2x%29%2B%288x%2B16%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29%2B%288x%2B16%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29%2B8%28x%2B2%29\"$ Factor out $\"8\"$ 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( " $\"%28x%2B8%29%28x%2B2%29\"$ Combine like terms. Or factor out the common term $\"x%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B10x%2B16\"$ factors to $\"%28x%2B8%29%28x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B10x%2B16=%28x%2B8%29%28x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B8%29%28x%2B2%29\"$ to get $\"x%5E2%2B10x%2B16\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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