document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=30217>Question 30217</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Explain why x to the 4th+2xsquared+4 has no real root while every polynomial function of degree 3 has at least 1 real root </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16908 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=16908\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Explain why x to the 4th+2xsquared+4 has no real root <BR>\n" );
document.write( "X^4+2X^2+4={(X^2)^2+2(X^2)(1)+(1)^2}-(1)^2+4 = (X^2+1)^2+3...OR...(X^2+1)^2=-3<BR>\n" );
document.write( "WE FIND THT THE SQUARE OF A NUMBER X^+1 IS NEGATIVE WHICH HAS NO REAL SOLUTION.(X^2+1)=i*SQRT.3<BR>\n" );
document.write( "HENCE <BR>\n" );
document.write( "X^2+1= i*SQRT.3<BR>\n" );
document.write( "X^2=-1+i*SQRT.3 = A COMPLEX NUMBER..HENCE ALL FOUR ROOTS OF X ARE COMPLEX/IMAGINARY NUMBERS.THAT IS THERE ARE NO REAL SOLUTIONS FOR X.<BR>\n" );
document.write( "IT IS POSSIBLE HERE SINCE THE TWO PAIRS OF COMPLEX ROOTS ARE CONJUGATE WITH THE RESULT THAT THE FUNCTIONS OF THEIR  SUM/PRODUCT IS REAL.SINCE THE COEFFICIENTS OF VARIABLE IN THE POLYNOMIAL ARE FUNCTIONS OF  SUM/PRODUCT OF THE ROOTS,IT IS NECESSARY THAT THE ROOTS SHOULD BE REAL OR CONJUGATE COMPLEX NUMBERS FOR THIS TO HAPPEN IN CASE OF A POLYNOMIAL WITH REAL COEFFICIENTS.IN THIS CASE WE HAVE <BR>\n" );
document.write( "IF A,A',B,B' ARE ROOTS (A AND A' ARE CONJUGATE ...B AND B' ARE CONJUGATE).. THEN<BR>\n" );
document.write( "A+A'+B+B'=0....POSSIBLE SINCE A+A' IS REAL AND B+B' IS REAL<BR>\n" );
document.write( "AA'+AB+AB'+A'B+A'B'+BB'=2...POSSIBLE SINCE AA' AND BB' ARE REAL AND <BR>\n" );
document.write( "AB+AB'+A'B+A'B'=A(B+B')+A'(B+B')=(A+A')(B+B')=REAL*REAL=REAL <BR>\n" );
document.write( ".....ETC...\r<BR>\n" );
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document.write( "while every polynomial function of degree 3 has at least 1 real root<BR>\n" );
document.write( "IN CASE OF 3 RD. DEGREE POLYNOMIAL ,WE HAVE 3 ROOTS..SO IF THEY ARE COMPLEX THEN AS WE SHOWED ABOVE FOR POLYNOMIAL WITH REAL COEFFICIENTS ..COMPLEX ROOTS OCCUR IN PAIRS OF CONJUGATE NUMBERS ONLY.SO IN 3 ROOTS 2 COULD BE COMPLEX CONJUGATES ,BUT THE 3 RD. HAS TO BE REAL IF THE COEFFICIENTS ARE REAL.\r<BR>\n" );
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