document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=174370>Question 174370</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the Rational Root Theorem to find all the roots of each equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4.   x'3 + 9x'2 + 19x – 4 = 0 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5.   2x'3 – x'2 + 10x – 5 = 0 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "6.   Two roots of a polynomial equation with real coefficients are 2 + 3i and {7}  ({} means square root) . Find two additional roots. Then find the degree of the polynomial.  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #129326 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=129326\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 4. x^3 + 9x^2 + 19x – 4 = 0 <BR>\n" );
document.write( "I graphed it and found a zero at x=-4<BR>\n" );
document.write( "Then using synthetic division you get:<BR>\n" );
document.write( "-4)....1....9....19....-4<BR>\n" );
document.write( "........1....5....-1...|..0\r<BR>\n" );
document.write( "\n" );
document.write( "Quotient: x^2+5x-1<BR>\n" );
document.write( "Use the quadratic formula to find the remaining zeroes:<BR>\n" );
document.write( "x = [-5 +- sqrt(25 -4*1*-1)]/2<BR>\n" );
document.write( "x = [-5 +- sqrt(29)]/2<BR>\n" );
document.write( "==================================\r<BR>\n" );
document.write( "\n" );
document.write( "5. 2x^3 – x^2 + 10x – 5 = 0 <BR>\n" );
document.write( "I graphed it and found a zero at x=1/2<BR>\n" );
document.write( "Then used synthetic division to get:<BR>\n" );
document.write( "1/2)....2....-1....10....-5<BR>\n" );
document.write( ".........2....0.....10...|..0\r<BR>\n" );
document.write( "\n" );
document.write( "Quotient: 2x^2 + 10<BR>\n" );
document.write( "Therefore the two other zeroes come from <BR>\n" );
document.write( "2x^2+10 = 0<BR>\n" );
document.write( "x^2 + 5 = 0<BR>\n" );
document.write( "x^2 = -5<BR>\n" );
document.write( "x = +/- isqrt(5)<BR>\n" );
document.write( "=====================\r<BR>\n" );
document.write( "\n" );
document.write( "6. Two roots of a polynomial equation with real coefficients are 2 + 3i and {7} ({} means square root) . Find two additional roots. Then find the degree of the polynomial.<BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "2-3i must be a zero if the coefficients are Real Numbers.<BR>\n" );
document.write( "-------------------------- <BR>\n" );
document.write( "Comment: Your problem seems to imply that sqrt(7) must be paired with -sqrt(7)<BR>\n" );
document.write( "But that is not true.<BR>\n" );
document.write( "--------------------------<BR>\n" );
document.write( "The answer to the problem you have posted is:<BR>\n" );
document.write( "The degree of the polynomial is three.<BR>\n" );
document.write( "---------------------<BR>\n" );
document.write( "Comment: If that 2nd zero you listed is really isqrt(7) then -isqrt(7)<BR>\n" );
document.write( "is a zero and the degree of the polynomial is four.<BR>\n" );
document.write( "=================================<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );