document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182855>Question 1182855</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The nature of the roots of the equation 3x^4 + 4x^3 + x - 1 = 0 is _____.\r<BR>\n" );
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document.write( "A. Three positive real roots and one negative real root.<BR>\n" );
document.write( "B. Three negative real roots and one positive real root.<BR>\n" );
document.write( "C. One negative real root and three complex roots.<BR>\n" );
document.write( "D. One positive real root, one negative real root and two complex roots.\r<BR>\n" );
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document.write( "--> Please help me on answering the correct nature of the roots on the equation and correct steps of doing it. Thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #812976 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=812976\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Use Descartes' Rule of Signs.<br><BR>\n" );
document.write( "(1) Put the polynomial in standard form -- that is, with decreasing powers of x.<BR>\n" );
document.write( "(2) The possible numbers of positive real roots are the number of sign changes in the terms of the polynomial, minus any multiple of 2.<BR>\n" );
document.write( "(3) The possible numbers of negative real roots are the number of sign changes in the terms of the polynomial when x is replaced by -x, minus any multiple of 2.<br><BR>\n" );
document.write( "For the polynomial in your problem....<br><BR>\n" );
document.write( "3x^4+4x^3+x-1<BR>\n" );
document.write( "The signs of the terms are +, +, +, -<BR>\n" );
document.write( "There is 1 sign change<BR>\n" );
document.write( "The only possible number of positive real roots is 1<br><BR>\n" );
document.write( "Replace x with -x in the polynomial; note that the signs of the coefficients of the even degree terms stay the same while the signs of the coefficients of the odd degree terms change.<BR>\n" );
document.write( "3x^4-4x^3-x-1<BR>\n" );
document.write( "The signs are +, -, -, -<BR>\n" );
document.write( "There is 1 sign change<BR>\n" );
document.write( "The only possible number of negative real roots is 1<br><BR>\n" );
document.write( "The polynomial is degree 4, so it has 4 roots.  There is only 1 positive real root and only 1 negative real root, so<br><BR>\n" );
document.write( "ANSWER: D. The polynomial has one positive real root, one negative real root, and one pair of complex roots.<br><BR>\n" );
document.write( "For a more interesting example of applying Descartes' Rule of Signs, consider this random polynomial:<br><BR>\n" );
document.write( "4x^8+3x^7-5x^6-3x^5-2x^3+x^2+3x-1<br><BR>\n" );
document.write( "The signs of the terms are +, +, -, -, -, +, +, -<BR>\n" );
document.write( "There are 3 sign changes<BR>\n" );
document.write( "The number of positive real roots is either 3 or 1<br><BR>\n" );
document.write( "Replace x with -x:<BR>\n" );
document.write( "4x^8-3x^7-5x^6+3x^5+2x^3+x^2-3x-1<BR>\n" );
document.write( "The signs of the terms are +, -, -, +, +, +, -, -<BR>\n" );
document.write( "There are 3 sign changes<BR>\n" );
document.write( "The number of negative real roots is either 3 or 1<br><BR>\n" );
document.write( "The nature of the roots is...<BR>\n" );
document.write( "either 1 or 3 positive real roots<BR>\n" );
document.write( "either 1 or 3 negative real roots<br><BR>\n" );
document.write( "The total number of real roots is then either 2, or 4, or 6; Since there are 8 roots, there are...<BR>\n" );
document.write( "either 6, or 4, or 2 complex roots<br><BR>\n" );
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