document.write( "Question 600655: x^2+8X+16 \n" ); document.write( "

Algebra.Com's Answer #379415 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor this.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B8x%2B16\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"8\"$ , 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 $\"8\"$ ?\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
\n" ); document.write( "\n" ); document.write( "1,2,4,8,16\r
\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 $\"8\"$ :\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 $\"4\"$ and $\"4\"$ add to $\"8\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"4\"$ and $\"4\"$ both multiply to $\"16\"$ and add to $\"8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"8x\"$ with $\"4x%2B4x\"$ . Remember, $\"4\"$ and $\"4\"$ add to $\"8\"$ . So this shows us that $\"4x%2B4x=8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%284x%2B4x%29%2B16\"$ Replace the second term $\"8x\"$ with $\"4x%2B4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B4x%29%2B%284x%2B16%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B4%29%2B%284x%2B16%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B4%29%2B4%28x%2B4%29\"$ Factor out $\"4\"$ 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( " $\"%28x%2B4%29%28x%2B4%29\"$ Combine like terms. Or factor out the common term $\"x%2B4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B4%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B8x%2B16\"$ factors to $\"%28x%2B4%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B8x%2B16=%28x%2B4%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B4%29%5E2\"$ to get $\"x%5E2%2B8x%2B16\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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