document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=950601>Question 950601</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use synthetic division to verify the upper and lower bounds of the real zeros of the polynomial p(x)= 2x^4-8x+3 upper:x=3, lower: x=-4\r<BR>\n" );
document.write( "\n" );
document.write( "B) p(x)=x^4-4x^3+15 upper: x=4, lower: x=-1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #580521 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=580521\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> polynomial is equal to 2x^4 - 8x + 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "upper bound is assumed to be x = 3<BR>\n" );
document.write( "lower bound is assumed to be x = -4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the test for upper and lower bounds of the real zeroes can be found at the following link.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut39_zero2.htm\"\" target = \"_blank\">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut39_zero2.htm</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "excerpt from that tutorial is shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Upper Bound<BR>\n" );
document.write( "If you divide a polynomial function f(x) by (x - c), where c > 0, using synthetic division and this yields all positive numbers, then c is an upper bound to the real roots of the equation f(x) = 0.\r<BR>\n" );
document.write( "\n" );
document.write( "Note that two things must occur for c to be an upper bound.  One is c > 0 or positive.  The other is that all the coefficients of the quotient as well as the remainder are positive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( "Lower Bound<BR>\n" );
document.write( "If you divide a polynomial function f(x) by (x - c), where c < 0, using synthetic division and this yields alternating signs, then c is a lower bound to the real roots of the equation f(x) = 0.  Special note that zeros can be either positive or negative. \r<BR>\n" );
document.write( "\n" );
document.write( "Note that two things must occur for c to be a lower bound.  One is c < 0 or negative.  The other is that successive coefficients of the quotient and the remainder have alternating signs.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you don't know how to do synthetic division, then check out the following tutorial:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/synthdiv2.htm\" target = \"_blank\">http://www.purplemath.com/modules/synthdiv2.htm</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "applying the upper bound test to 2x^4 - 8x + 3, you will get the following:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "first of all you have to replace the missing degrees.<BR>\n" );
document.write( "you also have to sort the equation in descending degree if it is not already sorted that way.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the equation of 2x^4 - 8x + 3 becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x^4 + 0x^3 + 0x^2 - 8x + 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if c is equal to 3, then x-c is equal to x-3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will be dividing 2x^4 + 0x^3 + 0x^2 - 8x + 3 by (x-3) using synthetic division.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "take the coefficients to form the dividend of the synthetic division.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2 + 0 + 0 - 8 + 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will be dividing 3 into 2 + 0 + 0 - 8 + 3 using synthetic division.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the answer will be 2 + 6 + 18 + 46 with a remainder of 141.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this represents the quotient of 2x^3 + 6x^2 + 18x + 46 + 141/(x-3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since all the coefficients are positive and the remainder is positive, then x = 3 is an upper bound of the real roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the lower bound test will have x = -4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if c is equal to -4, then x - c is equal to x - (-4) which is equal to x + 4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will be dividing 2x^4 + 0x^3 + 0x^2 - 8x + 3 by (x+4) using synthetic division.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "take the coefficients to form the dividend of the synthetic division.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2 + 0 + 0 - 8 + 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will be dividing -4 into 2 + 0 + 0 - 8 + 3 using synthetic division.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the answer will be 2 - 8 + 32 - 136 with a remainder of 547.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this represents the quotient of 2x^3 - 8x^2 + 32x - 136 + 547/(x+4)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since the signs of the quotient and the remainder are alternating between positive and negative, then x = -4 is a lower bound of the real roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the details of the calculations are shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/022502.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of the original equation of 2x^4 - 8x + 3 is shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/022501.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "notice that the roots are not at -4 and 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "they are between - 4 and 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in fact, the upper and lower bounds could have been more restrictive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "upper bound could easily have been 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "lower bound could easily have been -1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );