document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163519>Question 1163519</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question:  I always thought, till now, that the number of possible roots for a polynomial is equal(or atmost) to its highest degreee. Also I have been counting each radical and complex root as two since they all have +/- values (just like +/- n will be counted as two). Then I came across the following polynomials (degree four and five):<BR>\n" );
document.write( "     4X^4 +8X^3 -5X^2 -2X +1 = 0<BR>\n" );
document.write( "     3X^5 +2X^4 -15X^3 -10X^2 +12X +8 = 0 <BR>\n" );
document.write( "The second one as expected has 5 roots: +/-1,  +/-2, -2/3<BR>\n" );
document.write( "But the first one which (degree-4) has FIVE roots! viz., +/-1/2, +/-√2 and -1.\r<BR>\n" );
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document.write( "So which of assumption is wrong?<BR>\n" );
document.write( "   <BR>\n" );
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document.write( "What assumption?  There are no assumptions in the problem as stated.<br><BR>\n" );
document.write( "It is a fact that the number of roots of a polynomial is equal to the degree of the polynomial.<br><BR>\n" );
document.write( "What is not true is your statement that the 4th degree polynomial has 5 roots -- it of course does not.<br><BR>\n" );
document.write( "Specifically, -1 is not a root.<br><BR>\n" );
document.write( "The roots are +/-.5 and -1+/-sqrt(2).<br><BR>\n" );
document.write( "If you ever think you are finding a n-th degree polynomial with more than n roots, check the work you did to determine the roots, because something in your calculations has to be wrong.<br><BR>\n" );
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