document.write( "Question 1209702: Suppose P(x) is a polynomial of smallest possible degree such that:
\n" ); document.write( "* P(x) has rational coefficients.
\n" ); document.write( "* P(-2) = P(\sqrt{5}) = P(\sqrt{7}) = P(\sqrt{17}) = 0.
\n" ); document.write( "* P(-1) = 13.
\n" ); document.write( "Determine the value of P(0).
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Algebra.Com's Answer #849834 by ikleyn(52835) $\"\"$ $\"About$

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\n" ); document.write( "Suppose P(x) is a polynomial of smallest possible degree such that:
\n" ); document.write( "* P(x) has rational coefficients.
\n" ); document.write( "* P(-2) = P(sqrt{5}) = P(sqrt{7}) = P(sqrt{17}) = 0.
\n" ); document.write( "* P(-1) = 13.
\n" ); document.write( "Determine the value of P(0).
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document.write( "In this problem, a polynomial P(x) with rational coefficients has the roots\r\n" );
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document.write( "    -2, ,  ,  .\r\n" );
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document.write( "that are quadratic irrationalities.\r\n" );
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document.write( "To make a polynomial with rational coefficients, which has these roots, we shall\r\n" );
document.write( "associate the conjugate number over Q with every quadratic irrationality.\r\n" );
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document.write( "So, P(x) has the form\r\n" );
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document.write( "    P(x) =  = \r\n" );
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document.write( "         = .\r\n" );
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document.write( "Here \"a\" is the coefficient, now unknown.  We will determine it from the condition P(-1) = 13.\r\n" );
document.write( "It gives\r\n" );
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document.write( "            = 13,  \r\n" );
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document.write( "or\r\n" );
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document.write( "           a*1*(-4)*(-6)*(-16) = 13,\r\n" );
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document.write( "           a*(-384) = 13,\r\n" );
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document.write( "           a = .\r\n" );
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document.write( "Now the polynomial is fully defined  P(x) = .\r\n" );
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document.write( "To find P(0), plug in 0 instead of x:  \r\n" );
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document.write( "    P(0) =  =  =  = .\r\n" );
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document.write( "ANSWER.  P(x) =   and  P(0) = .\r\n" );
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