document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=216529>Question 216529</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I need some help with my American School Algebra 2 work. Given X^*3-4x^*2+2x+1=0 <BR>\n" );
document.write( "A. How many possible positive roots are there?<BR>\n" );
document.write( "B. How many possible negative roots are there?<BR>\n" );
document.write( "C. What are the possible rational roots?<BR>\n" );
document.write( "D. Using synthetic substitution, which of the possible rational roots is actually a root of the equation?<BR>\n" );
document.write( "E. Find the irrational roots of the equation(hint:ust the quadratic formula to solve the depressed equation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #163501 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=163501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Hi, I need some help with my American School Algebra 2 work. <BR>\n" );
document.write( "Given x^3-4x^2+2x+1=0 <BR>\n" );
document.write( "A. How many possible positive roots are there?<BR>\n" );
document.write( "Using Decartes' rule, there are either 2 or 0 positive roots.<BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "B. How many possible negative roots are there?<BR>\n" );
document.write( "Descartes again, 1 negative root.<BR>\n" );
document.write( "You can find Descartes on google.<BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "C. What are the possible rational roots?<BR>\n" );
document.write( "+1 is a root.  I found it by graphing the function.<BR>\n" );
document.write( "Then dividing it out, you're left with<BR>\n" );
document.write( "x^2-3x-1 = 0<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-3x%2B-1+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-3x%2B-1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-3%29%5E2-4%2A1%2A-1=13 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-3%29%5E2-4%2A1%2A-1=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=13 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--3%2B-sqrt%28+13+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--3%2B-sqrt%28+13+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+13+%29%29%2F2%5C1+=+3.30277563773199 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+13+%29%29%2F2%5C1+=+3.30277563773199\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-3%29-sqrt%28+13+%29%29%2F2%5C1+=+-0.302775637731995 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-3%29-sqrt%28+13+%29%29%2F2%5C1+=+-0.302775637731995\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-3x%2B-1 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-3x%2B-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-3x%2B-1+=+%28x-3.30277563773199%29%2A%28x--0.302775637731995%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-3x%2B-1+=+%28x-3.30277563773199%29%2A%28x--0.302775637731995%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 3.30277563773199, -0.302775637731995.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-3%2Ax%2B-1+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-3%2Ax%2B-1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "-----------<BR>\n" );
document.write( "x = (3+sqrt(13))/2<BR>\n" );
document.write( "x = (3-sqrt(13))/2<BR>\n" );
document.write( "---------------------\r<BR>\n" );
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document.write( "D. Using synthetic substitution, which of the possible rational roots is actually a root of the equation?<BR>\n" );
document.write( "E. Find the irrational roots of the equation(hint:ust the quadratic formula to solve the depressed equation.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );