document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=32880>Question 32880</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the polynomial p(x)= x^4 + 2x^2 + 2x + 2 cannot be expressed as the product of two quadratic polynomials of form x^2 + ax + b and x^2 + cx +d where a,b,c, and d are any integers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #19332 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=19332\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> F(X)=X^+2X^2+2X+2.....IF THIS HAS TO BE PRODUCT OF 2 QUADRATICS THEN<BR>\n" );
document.write( "THEY HAVE TO BE<BR>\n" );
document.write( "X^2+AX+1 & X^2-AX+2...OR......<BR>\n" );
document.write( "X^2+AX-1 & X^2-AX-2.....AS CONSTANT TERM =2 .ITS INTEGRAL FACTORS ARE<BR>\n" );
document.write( "1 &2 OR -1 & -2.<BR>\n" );
document.write( "FURTHER COEFFICIENT OF X^3=0 .,SO IF THERE IS AX IN ONE FACTOR,THERE<BR>\n" );
document.write( "SHOULD BE -AX IN ANOTHER FACTOR.<BR>\n" );
document.write( "THIS LEAVES US....... BY EQUATING COEFFICIENTS OF X AND X^2 TERMS...<BR>\n" );
document.write( "(X^2+AX+1)(X^2-AX+2)=X^4-AX^3+2X^2+AX^3-(A^2)(x^2)+2AX+X^2-AX+2<BR>\n" );
document.write( "=X^4+X^2(3-A^2)+AX+2=X^4+2X^2+2X+2...SO<BR>\n" );
document.write( "3-A^2=2....AND.....A=2.....WHICH ARE NOT CONSISTENT.HENCE THERE IS NO<BR>\n" );
document.write( "SOLUTION....SIMILARLY...WE CAN PROVE THIS FOR<BR>\n" );
document.write( "(X^2+AX-1)(X^2-AX-2)=X^4-AX^3-2X^2+AX^3-(A^2)(x^2)-2AX-X^2+AX+2<BR>\n" );
document.write( "=X^4-X^2(3+A^2)-AX+2=X^4+2X^2+2X+2...SO<BR>\n" );
document.write( "3+A^2=2....AND.....A= -2.....WHICH ARE NOT CONSISTENT.HENCE THERE IS<BR>\n" );
document.write( "NO SOLUTION....<BR>\n" );
document.write( "HENCE F(X) IS NOT REDUCIBLE OR CAN NOT BE FACTORISED INTO 2 QUADRATIC<BR>\n" );
document.write( "EXPRESSIONS. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );