document.write( "Question 238740: Factor
\n" ); document.write( "(a+4)^2 - 2(a+4) +1 \n" ); document.write( "

Algebra.Com's Answer #175464 by jim_thompson5910(35256) $\"\"$ $\"About$

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Let $\"z=a%2B4\"$ . So the expression $\"%28a%2B4%29%5E2+-+2%28a%2B4%29+%2B1\"$ then becomes $\"z%5E2-2z%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"z%5E2-2z%2B1\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"1\"$ to get $\"%281%29%281%29=1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"1\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?\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 $\"1\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"1\"$ :\r
\n" ); document.write( "\n" ); document.write( "1\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 $\"1\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*1 = 1
\n" ); document.write( "(-1)*(-1) = 1\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 $\"-2\"$ :\r
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First Number	Second Number	Sum
1	1	1+1=2
-1	-1	-1+(-1)=-2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"-1\"$ add to $\"-2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"-1\"$ both multiply to $\"1\"$ and add to $\"-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-2z\"$ with $\"-z-z\"$ . Remember, $\"-1\"$ and $\"-1\"$ add to $\"-2\"$ . So this shows us that $\"-z-z=-2z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"z%5E2%2Bhighlight%28-z-z%29%2B1\"$ Replace the second term $\"-2z\"$ with $\"-z-z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28z%5E2-z%29%2B%28-z%2B1%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z-1%29%2B%28-z%2B1%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z-1%29-1%28z-1%29\"$ Factor out $\"1\"$ 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( " $\"%28z-1%29%28z-1%29\"$ Combine like terms. Or factor out the common term $\"z-1\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28z-1%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"z%5E2-2z%2B1\"$ factors to $\"%28z-1%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"z%5E2-2z%2B1=%28z-1%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now plug in $\"z=a%2B4\"$ to go from $\"%28z-1%29%5E2\"$ to $\"%28a%2B4-1%29%5E2\"$ . Now simplify to get $\"%28a%2B3%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"%28a%2B4%29%5E2+-+2%28a%2B4%29+%2B1\"$ factors to $\"%28a%2B3%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"%28a%2B4%29%5E2+-+2%28a%2B4%29+%2B1=%28a%2B3%29%5E2\"$ .\r
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