document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=189408>Question 189408</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that 7 + &#8730;2 is an algebraic number </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142195 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=142195\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "An algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients.\r<BR>\n" );
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document.write( "So, if you can show that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 7 + sqrt{2} \"> is the root of a non-zero polynomial with rational coefficients, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 7 + sqrt{2} \"> must be an algebraic number.\r<BR>\n" );
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document.write( "Irrational roots of quadratic polynomials always come in conjugate pairs, that is if <i>a</i> is rational, <i>b</i> is irrational, and <i>a</i> + <i>b</i> is a root of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha_1x^2 + \alpha_2x + \alpha_3\ =\ 0\"> then <i>a</i> - <i>b</i> must also be a root of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha_1x^2 + \alpha_2x + \alpha_3\ =\ 0\">.\r<BR>\n" );
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document.write( "So, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = 7 + sqrt{2} \"> is a root, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = 7 - sqrt{2}\"> must be the other root, and the factors of the desired polynomial, if it exists, must be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  (x - (7 + sqrt{2}))(x - (7 - sqrt{2}))\">\r<BR>\n" );
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document.write( "Applying FOIL:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2 - (7 - sqrt{2})x - (7 + sqrt{2})x + (49 - 2)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 14x + 47\">\r<BR>\n" );
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document.write( "which is a non-zero polynomial in one variable with integer, and therefore rational, coefficients, and therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 7 + sqrt{2} \"> is an algebraic number.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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