document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201181>Question 1201181</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use Descartes rule of signs to find the number of real zeros possible. \r<BR>\n" );
document.write( "\n" );
document.write( "P(x) = x^5 + 4x^3 − x^2 + 8x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835446 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=835446\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "It's fairly clear that x = 0 is a root of P(x)<BR>\n" );
document.write( "This is because P(x) = 0 when x = 0<BR>\n" );
document.write( "P(x) = x^5+4x^3-x^2+8x<BR>\n" );
document.write( "P(0) = (0)^5+4(0)^3-(0)^2+8(0)<BR>\n" );
document.write( "P(0) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's factor x from each term<BR>\n" );
document.write( "P(x) = x^5+4x^3-x^2+8x<BR>\n" );
document.write( "P(x) = x(x^4+4x^2-x+8)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We have something of the form<BR>\n" );
document.write( "P(x) = x*Q(x)<BR>\n" );
document.write( "where<BR>\n" );
document.write( "Q(x) = x^4+4x^2-x+8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's use Descartes rule of signs to determine the possible number of roots for Q(x).<BR>\n" );
document.write( "It will in turn help us determine the possible number of roots for P(x).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We need to count the number of sign changes in Q(x) to determine the possible number of positive roots for Q(x).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Think of x^4 as +x^4<BR>\n" );
document.write( "The jump from +x^4 to +4x^2 has no sign change because each term is positive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But the jump from +4x^2 to -x has the sign go from positive to negative. We have a sign change.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another sign change happens when going from -x to +8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Q(x) has 2 sign changes in total.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28300%2C300%2C-5%2C5%2C-3%2C7%2C%0D%0Ablue%28%0D%0Aarc%281.8%2C2%2C1%2C1%2C180%2C360%29%2C%0D%0Alocate%281.2%2C3%2Cmatrix%281%2C2%2Csign%2Cchange%29%29%2C%0D%0Aline%282.25%2C1.9%2C2.1375%2C2.0949%29%2Cline%282.25%2C1.9%2C2.3625%2C2.0949%29%2C%0D%0Alocate%28-5%2C-2%2Cmatrix%281%2C5%2Cblue%2C%22=%22%2Cnegative%2Cto%2Cpositive%29%29%0D%0A%29%2C%0D%0A%0D%0Alocate%28-2%2C2%2Cmatrix%281%2C3%2CQ%28x%29%2C+%22=%22%2C+x%5E4+%2B+4x%5E2+-+x+%2B+8%29%29%2C%0D%0A%0D%0Ared%28%0D%0Aarc%280.8%2C1.2%2C1%2C1%2C0%2C180%29%2C%0D%0Alocate%28-1%2C0.8%2Cmatrix%281%2C2%2Csign%2Cchange%29%29%2C%0D%0Aline%281.275%2C1.25%2C1.3875%2C1.0551%29%2Cline%281.275%2C1.25%2C1.1625%2C1.0551%29%2C%0D%0Alocate%28-5%2C-1%2Cmatrix%281%2C5%2Cred%2C%22=%22%2Cpositive%2Cto%2Cnegative%29%29%0D%0A%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It means Q(x) has either 2 positive real roots or 0 positive real roots.<BR>\n" );
document.write( "We step down by 2 because complex roots (of the form a+bi) come in conjugate pairs. <BR>\n" );
document.write( "This is only when all coefficients of the polynomial are real numbers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check out this example<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://andymath.com/descartes-rule-of-signs/\">https://andymath.com/descartes-rule-of-signs/</a><BR>\n" );
document.write( "and this video may help<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.youtube.com/watch?v=YaU5JTe3cPU\">https://www.youtube.com/watch?v=YaU5JTe3cPU</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now we need to look at the sign changes of Q(-x) to determine the possible number of negative roots for Q(x).<BR>\n" );
document.write( "Replace each x with -x and simplify.<BR>\n" );
document.write( "Q(x) = x^4 + 4x^2 - x + 8<BR>\n" );
document.write( "Q(-x) = (-x)^4 + 4(-x)^2 - (-x) + 8<BR>\n" );
document.write( "Q(-x) = x^4 + 4x^2 + x + 8<BR>\n" );
document.write( "There are NO sign changes at all here.<BR>\n" );
document.write( "Each term is positive.<BR>\n" );
document.write( "Therefore, the number of possible negative real roots is 0. <BR>\n" );
document.write( "In other words, Q(x) will not have x-intercepts in the negative territory.<BR>\n" );
document.write( "Furthermore, P(x) doesn't have any x-intercepts in negative territory.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here are all the possible outcomes for Q(x)<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Positive</TD><TD >Negative</TD><TD >Imaginary</TD><TD >Total</TD></tr><tr><TD >2</TD><TD >0</TD><TD >2</TD><TD >4</TD></tr><tr><TD >0</TD><TD >0</TD><TD >4</TD><TD >4</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "The first row represents having 2 positive real number roots. Notice the total is always 4 due to the Fundamental Theorem of Algebra. <BR>\n" );
document.write( "This theorem says the total number of roots (real+imaginary) must be the degree of the polynomial.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So if we know there are 2 positive roots and 0 negative roots. Then so far we have 2+0 = 2 real roots. The remaining 4-2 = 2 roots must be in the form a+bi where i = sqrt(-1)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We use the same logic for the second row as well. All four roots for that row are complex.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is what the table looks like for Q(x) when we list the possible number of real roots and imaginary roots.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Real</TD><TD >Imaginary</TD><TD >Total</TD></tr><tr><TD >2</TD><TD >2</TD><TD >4</TD></tr><tr><TD >0</TD><TD >4</TD><TD >4</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is what the table looks like for P(x) when we list the possible number of real roots and imaginary roots.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Real</TD><TD >Imaginary</TD><TD >Total</TD></tr><tr><TD ><font color=red>3</font></TD><TD >2</TD><TD >5</TD></tr><tr><TD ><font color=red>1</font></TD><TD >4</TD><TD >5</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "I've added 1 to each real count for Q(x) since we determined earlier x = 0 was a root of P(x).<BR>\n" );
document.write( "Each total also increases by 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: <font color=red>Either 3 real roots or 1 real root.</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );