document.write( "Question 212101: Dear Tutor,\r
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\n" ); document.write( "\n" ); document.write( "I would like to know how can I arrive/make:
\n" ); document.write( "y*z*(z-y)+x*y*(y-x)+x*z*(x-z) \r
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\n" ); document.write( "\n" ); document.write( "into\r
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\n" ); document.write( "\n" ); document.write( "(x-y)*(y-z)*(z-x)\r
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\n" ); document.write( "\n" ); document.write( "I have tried multiplying all factors also tried to divide by factor:y and than factor:z but that's where I got stuck so I'm not sure if I'm even close to the right method.\r
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\n" ); document.write( "\n" ); document.write( "Thank you for your time and effort,\r
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\n" ); document.write( "\n" ); document.write( "Gabi \n" ); document.write( "

Algebra.Com's Answer #160183 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"y%2Az%2A%28z-y%29%2Bx%2Ay%2A%28y-x%29%2Bx%2Az%2A%28x-z%29\"$
\n" ); document.write( "This is a tough one. For one, you have to be good at factoring by grouping. Another key is recognizing that:
\n" ); document.write( "z-y = -1(y-z)
\n" ); document.write( "y-x = -1(x-y)
\n" ); document.write( "x-z = -1(z-x)
\n" ); document.write( "Substituting the first of these into the original we get:
\n" ); document.write( " $\"-y%2Az%2A%28y-z%29%2Bx%2Ay%2A%28y-x%29%2Bx%2Az%2A%28x-z%29\"$
\n" ); document.write( "Now we have one of the desired factors, (y-z), present. The rest is manipulations with the goal of having (y-z) as a common factor. First we'll multiply out the reset of the expression:
\n" ); document.write( " $\"-y%2Az%2A%28y-z%29%2Bx%2Ay%5E2+-+x%5E2%2Ay+%2B+x%5E2%2Az+-+xz%5E2\"$
\n" ); document.write( "Rearranging the terms for factoring by grouping we get:
\n" ); document.write( " $\"-y%2Az%2A%28y-z%29%2B+red%28x%2Ay%5E2+-+xz%5E2%29+%2B+green%28-x%5E2%2Ay+%2B+x%5E2%2Az%29\"$
\n" ); document.write( "Factoring x from the \"red\" terms and $\"-x%5E2\"$ from the \"green\" terms we get:
\n" ); document.write( " $\"-y%2Az%2A%28y-z%29%2B+x%2Ared%28y%5E2+-+z%5E2%29+%2B+%28-x%5E2%29%2A%28y+-+z%29\"$
\n" ); document.write( "Factoring the \"red\" expression as a difference of squares we get:
\n" ); document.write( " $\"-y%2Az%2A%28y-z%29+%2B+x%2A%28y%2Bz%29%28y-z%29+%2B+%28-x%5E2%29%2A%28y+-+z%29\"$
\n" ); document.write( "Now we have (y-z) as a common factor of each term. We can factor it out:
\n" ); document.write( " $\"%28y-z%29%28-y%2Az+%2B+red%28x%2A%28y%2Bz%29%29+-+x%5E2%29\"$
\n" ); document.write( "Multiplying out the \"red\" expression we get:
\n" ); document.write( " $\"%28y-z%29%28-y%2Az+%2B+xy+%2B+xz+-+x%5E2%29\"$
\n" ); document.write( "Rearranging the terms for factoring by grouping we get:
\n" ); document.write( " $\"%28y-z%29%28red%28xz+-+y%2Az%29+%2B+green%28-x%5E2+%2B+xy%29%29\"$
\n" ); document.write( "Factoring z from the \"red\" terms and -x from the \"green\" terms we get:
\n" ); document.write( " $\"%28y-z%29%28z%28x+-+y%29+%2B+%28-x%29%28x+-+y%29%29\"$
\n" ); document.write( "Now we can factor the (x-y)'s:
\n" ); document.write( " $\"%28y-z%29%28%28x-y%29%28z-x%29%29\"$
\n" ); document.write( "With the Associative Property we can remove the extra parentheses giving:
\n" ); document.write( " $\"%28y-z%29%28x-y%29%28z-x%29\"$
\n" ); document.write( "And with the Commutative Property we have:
\n" ); document.write( " $\"%28x-y%29%28y-z%29%28z-x%29\"$ \r
\n" ); document.write( "\n" ); document.write( "and we're done! \n" ); document.write( "

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