document.write( "Question 367535: Factor completely 2x^4 + x^3 - x^2 -x \n" ); document.write( "

Algebra.Com's Answer #262034 by Jk22(389) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"2x%5E4+%2B+x%5E3+-+x%5E2+-x\"$
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\n" ); document.write( " $\"+x%282x%5E3+%2B+x%5E2+-+x+-+1%29\"$
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\n" ); document.write( "the polynomial inside is factorizable over R :
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\n" ); document.write( "2x^3 +
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\n" ); document.write( " $\"2x+%28+%28x+%2B+1%2F6%29%5E3+-+x%2F12+-+x%2F2+-+1%2F2+-+1%2F216%29\"$
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\n" ); document.write( " $\"2x+%28+%28x+%2B+1%2F6%29%5E3+-+7x%2F12+-+109%2F216+%29+\"$ \r
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\n" ); document.write( "\n" ); document.write( "let : $\"x+%2B+1%2F6+=+y\"$ , we get in the parenthesis :
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\n" ); document.write( " $\"y%5E3+-+%287%2F12%29%2Ay+-+109%2F216+%2B+7%2F12+=+y%5E3+-+%287%2F12%29%2Ay+%2B+27%2F216\"$
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\n" ); document.write( "\n" ); document.write( "writing $\"y+=+p+%2B+q\"$ ,
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\n" ); document.write( " $\"p%5E3+%2B+q%5E3+%2B+3pq+%28p%2Bq%29+-+%287%2F12%29%2A%28p%2Bq%29+%2B+27%2F216+=+0\"$
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\n" ); document.write( "we get :
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\n" ); document.write( " $\"p%5E3+%2B+q%5E3+=+-27%2F216\"$
\n" ); document.write( " $\"3pq+=+7%2F12\"$ \r
\n" ); document.write( "\n" ); document.write( "the latter gives : $\"q+=+7%2F36%2A1%2Fp\"$ , if substituted in the equation before :
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\n" ); document.write( " $\"p%5E3+%2B+%287%2F36%29%5E3%2Fp%5E3+=+-27%2F216\"$ times $\"p%5E3\"$
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\n" ); document.write( " $\"p%5E6+%2B+%2827%2F216%29%2Ap%5E3+%2B+%287%2F36%29%5E3+=+0\"$ \r
\n" ); document.write( "\n" ); document.write( "with 2nd degree we find : \r
\n" ); document.write( "\n" ); document.write( " $\"p%5E3+=+%28-27%2F216+%2B-+sqrt%28%2827%2F216%29%5E2+-+4%2A%287%2F36%29%5E3%29%29%2F2\"$
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\n" ); document.write( " $\"q+=+7%2F36%2A1%2Fp\"$ and $\"x++%2B+1%2F6+=+p+%2B+7%2F36%2A1%2Fp\"$
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\n" ); document.write( "with

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\n" ); document.write( "hence the solution is

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\n" ); document.write( "where 2 roots are identical.
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\n" ); document.write( "we can write the polynomial as : 2x(x-x[++])(x-x[+-])(x-x[--])
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