document.write( "Question 1160401: If 1 and w are two of the five roots of w^5=1, then show that 1+w+w^2+w^3+w^4 = 0 \n" ); document.write( "

Algebra.Com's Answer #783698 by ikleyn(52810) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " I will show you two ways to solve this problem.\r
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\n" ); document.write( "\n" ); document.write( "Proof 1.\r
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document.write( "There is an identity, which is valid for any real or complex number w\r\n" );
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document.write( "     =       (1)\r\n" );
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document.write( "You may prove it directly, by making FOIL.\r\n" );
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document.write( "OR you may know it from the formula for the sum of a geometric progression.\r\n" );
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document.write( "In any way, I assume you know it.\r\n" );
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document.write( "Next step.  If w is the root of the equation   = 1, then the left side in (1) becomes equal to zero,\r\n" );
document.write( "and you get\r\n" );
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document.write( "     = 0.     (2)\r\n" );
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document.write( "Since w is different from 1, the factor (w-1) is not zero, and you can cancel it in both sides of (2).\r\n" );
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document.write( "You will get then\r\n" );
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document.write( "     = 0,\r\n" );
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document.write( "exactly what should be proved.\r\n" );
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document.write( "Thus the proof is completed.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Proof 2\r
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document.write( "If w is the root of the equation w^5 = 1,  then 1, w, w^2, w^3, w^4 is the set of ALL 5 (five) roots of this equation.\r\n" );
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document.write( "    I proved it for you in my PREVIOUS post.\r\n" );
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document.write( "Now apply the Vieta's theorem: \r\n" );
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document.write( "    for any polynomial equation of the degree n with the leading coefficient 1, \r\n" );
document.write( "    the sum of its roots is equal to the coefficient at  taken with the opposite sign.\r\n" );
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document.write( "Since in the given equation  x^5-1 = 0  the coefficient at    is 0 (zero, ZERO),\r\n" );
document.write( "the sum of its roots is equal to zero:\r\n" );
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document.write( "    1 + w + w^2 + w^3 + w^4 = 0.\r\n" );
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document.write( "It is exactly what has to be proved.\r\n" );
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document.write( "The proof is completed.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus you have two (TWO) proofs, to your great satisfaction.\r
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