document.write( "Question 1189226: Please explain how to solve this: https://docs.google.com/document/d/1jpqO35zDSlUDFyoB_pZ0TZF8JsOYXjmlSxNh7r31O-s/edit?usp=sharing \n" ); document.write( "

Algebra.Com's Answer #820555 by ikleyn(52814) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Consider the polynomial P(z) = z^5 - 10z^2 + 15z - 6 of complex variable z.\r
\n" ); document.write( "\n" ); document.write( "The polynomial can be written in the form P(z) = (z-1)^3*(z^2 + bz+ c).\r
\n" ); document.write( "\n" ); document.write( "Consider the function q(x) = x^5 - 10x^2 + 15x - 6, for real x.\r
\n" ); document.write( "\n" ); document.write( "a. Write down the sum and the product of the roots of P(z).
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\n" ); document.write( "\n" ); document.write( "            The answer in the post by @Boreal is INCORRECT.\r
\n" ); document.write( "\n" ); document.write( "            I came to bring a correct solution with correct answer.\r
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\n" ); document.write( "\n" ); document.write( "            Notice that the problem is too wordy.\r
\n" ); document.write( "\n" ); document.write( "            Its 1st line combined with the 4th line is totally enough to present a correct formulation.\r
\n" ); document.write( "\n" ); document.write( "            The 2-nd line and the 3rd line are unnecessary and excessive.\r
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document.write( "The most interesting fact is that to get the answer, you  do not need  find the roots \r\n" );
document.write( "and/or factorize the polynomial in an explicit form.\r\n" );
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document.write( "It is enough to apply the Vieta's theorem.  It says that the sum of the roots of the given polynomial equals \r\n" );
document.write( "to the coefficient at x^4, taken with the opposite sign.\r\n" );
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document.write( "In our case, the coefficient at x^4 is 0 (zero, ZERO); so, the sum of the roots equals 0 (zero, ZERO).\r\n" );
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document.write( "Regarding the product of the roots, apply the Vieta's theorem again.  It says that the product of the roots \r\n" );
document.write( "of the given polynomial equals to the constant term, taken with the opposite sign.\r\n" );
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document.write( "In our case, the constant term is -6;  hence, the product of the roots is 6.\r\n" );
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document.write( "ANSWER.  The sum of the roots is 0 (zero, ZERO).  The product of the roots is 6.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved and thoroughly/carefully/comprehensively explained.\r
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\n" ); document.write( "\n" ); document.write( "On Vieta's theorem, see this link\r
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\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Vieta%27s_formulas\r
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\n" ); document.write( "\n" ); document.write( "The solution by @Boreal has arithmetic mistakes, that lead to incorrect answer.\r
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