document.write( "Question 610313: Factor 64a^2-16ab+b^2 \n" ); document.write( "

Algebra.Com's Answer #384292 by jim_thompson5910(35256) $\"\"$ $\"About$

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Looking at the expression $\"64a%5E2-16ab%2Bb%5E2\"$ , we can see that the first coefficient is $\"64\"$ , the second coefficient is $\"-16\"$ , and the last coefficient is $\"1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"64\"$ by the last coefficient $\"1\"$ to get $\"%2864%29%281%29=64\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"64\"$ (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 $\"64\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"64\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8,-16,-32,-64\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 $\"64\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*64 = 64
\n" ); document.write( "2*32 = 64
\n" ); document.write( "4*16 = 64
\n" ); document.write( "8*8 = 64
\n" ); document.write( "(-1)*(-64) = 64
\n" ); document.write( "(-2)*(-32) = 64
\n" ); document.write( "(-4)*(-16) = 64
\n" ); document.write( "(-8)*(-8) = 64\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	64	1+64=65
2	32	2+32=34
4	16	4+16=20
8	8	8+8=16
-1	-64	-1+(-64)=-65
-2	-32	-2+(-32)=-34
-4	-16	-4+(-16)=-20
-8	-8	-8+(-8)=-16

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-8\"$ and $\"-8\"$ add to $\"-16\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-8\"$ and $\"-8\"$ both multiply to $\"64\"$ and add to $\"-16\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-16ab\"$ with $\"-8ab-8ab\"$ . Remember, $\"-8\"$ and $\"-8\"$ add to $\"-16\"$ . So this shows us that $\"-8ab-8ab=-16ab\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"64a%5E2%2Bhighlight%28-8ab-8ab%29%2Bb%5E2\"$ Replace the second term $\"-16ab\"$ with $\"-8ab-8ab\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2864a%5E2-8ab%29%2B%28-8ab%2Bb%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"8a%288a-b%29%2B%28-8ab%2Bb%5E2%29\"$ Factor out the GCF $\"8a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"8a%288a-b%29-b%288a-b%29\"$ Factor out the $\"-b\"$ from the second group.\r
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\n" ); document.write( "\n" ); document.write( " $\"%288a-b%29%288a-b%29\"$ Factor out $\"8a-b\"$ from the entire expression.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"64a%5E2-16ab%2Bb%5E2\"$ factors to $\"%288a-b%29%288a-b%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"64a%5E2-16ab%2Bb%5E2=%288a-b%29%288a-b%29\"$ for all values of 'a' and 'b'.\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%288a-b%29%288a-b%29\"$ to get $\"64a%5E2-16ab%2Bb%5E2\"$ back again or by graphing the original expression and the answer (the two graphs should be identical).
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