document.write( "Question 133324: Rewrite the expression x^3 - 2x^2 y + x y^2 so that it is easy to calculate the its value without a calculator when x= 21 and y = 19 \n" ); document.write( "

Algebra.Com's Answer #97518 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"x%5E3-2x%5E2y%2B1xy%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%5E2-2xy%2By%5E2%29\"$ Factor out the GCF $\"x\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2-2xy%2By%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"1x%5E2-2xy%2B1y%5E2\"$ we can see that the first term is $\"1x%5E2\"$ and the last term is $\"1y%5E2\"$ where the coefficients are 1 and 1 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the middle coefficient -2? Let's list all of the factors of 1:\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( "-1 ...List the negative factors as well. 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\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-1)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -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 this list we can see that -1 and -1 add up to -2 and multiply to 1\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"1x%5E2-2xy%2B1y%5E2\"$ , replace $\"-2xy\"$ with $\"-1xy%2B-1xy\"$ (notice $\"-1xy%2B-1xy\"$ adds up to $\"-2xy\"$ . So it is equivalent to $\"-2xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"1x%5E2%2Bhighlight%28-1xy%2B-1xy%29%2B1y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1x%5E2-1xy-1xy%2B1y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%281x%5E2-1xy%29%2B%28-1xy%2B1y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-y%29-y%28x-y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-y%29%28x-y%29\"$ Since we have a common term of $\"x-y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"1x%5E2-1xy-1xy%2B1y%5E2\"$ factors to $\"%28x-y%29%28x-y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"1x%5E2-2xy%2B1y%5E2\"$ factors to $\"%28x-y%29%28x-y%29\"$ (since $\"1x%5E2-2xy%2B1y%5E2\"$ is equivalent to $\"1x%5E2-1xy-1xy%2B1y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%28x-y%29%28x-y%29\"$ is equivalent to $\"%28x-y%29%5E2\"$ since the term $\"x-y\"$ occurs twice. So $\"1x%5E2-2xy%2B1y%5E2\"$ also factors to $\"%28x-y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"x%28x%5E2-2xy%2By%5E2%29\"$ and factors further to $\"x%28x-y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E3-2x%5E2y%2Bxy%5E2\"$ factors to $\"x%28x-y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"21%2821-19%29%5E2\"$ Plug in x= 21 and y = 19 \r
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\n" ); document.write( "\n" ); document.write( " $\"21%282%29%5E2\"$ Subtract \r
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\n" ); document.write( "\n" ); document.write( " $\"21%284%29\"$ Square 2 to get 4\r
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\n" ); document.write( "\n" ); document.write( " $\"84\"$ Multiply \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E3+-+2x%5E2+y+%2B+x+y%5E2=84\"$ when x= 21 and y = 19
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