document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=176615>Question 176615</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given x^3-4x^2+2x+1=0\r<BR>\n" );
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document.write( "A. How many possible negative roots are there?\r<BR>\n" );
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document.write( "B. Using synthetic substitution, which of the possible rational roots is actually a root of the equation? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #131693 by </B><A HREF=/tutors/aboutme.mpl?userid=Mathtut><B>Mathtut(3670)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Mathtut><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=131693\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A) you must use f(-x) to find out the number of negative root possibles....f(-x)=-x^3-4x^2-2x+1=0<BR>\n" );
document.write( "number of negative real roots is less than or equal to the number of variations in the function f (- x). <BR>\n" );
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document.write( "there is 1 variation in f(-x) therefore there is one negative real root<BR>\n" );
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document.write( "1 or -1 could be roots because they are factors of the constant which is 1<BR>\n" );
document.write( ":<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-4x%5E2%2B2x%2B1=0 ALIGN=MIDDLE  ALT=\"x%5E3-4x%5E2%2B2x%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E3-4%281%29%5E2%2B2%281%29%2B1=0 ALIGN=MIDDLE  ALT=\"1%5E3-4%281%29%5E2%2B2%281%29%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > therefore 1 is a root<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%5E3-4%28-1%29%5E2%2B2%28-1%29%2B1=-8 ALIGN=MIDDLE  ALT=\"%28-1%29%5E3-4%28-1%29%5E2%2B2%28-1%29%2B1=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > therefore -1 is NOT a root<BR>\n" );
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document.write( "This relationship is always true: If a polynomial has rational roots, then those roots will be fractions of the form (plus-or-minus) (factor of the constant term) / (factor of the leading coefficient). However, not all fractions of this form are necessarily zeroes of the polynomial. Indeed, it may happen that none of the fractions so formed is actually a zero of the polynomial.\r<BR>\n" );
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document.write( "since 1 is our constant and 1 is our leading coefficient the only possible rational roots are 1 divided by 1 or -1 which is the same as those terms we have already tested.<BR>\n" );
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document.write( "now synthetic division using the root 1 (x-1)<BR>\n" );
document.write( ":<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
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document.write( "   |\r\n" );
document.write( "  1|1 -4  2   1\r\n" );
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document.write( "    1 -3 -1   0\r\n" );
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document.write( "so when we factor out the root 1 we get<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3x-1=0 ALIGN=MIDDLE  ALT=\"x%5E2-3x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( ":and using the quadratic formula we get 2 irrational roots from this depressed equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2%2Bsqrt%2813%29%2F2 ALIGN=MIDDLE  ALT=\"3%2F2%2Bsqrt%2813%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2-sqrt%2813%29%2F2 ALIGN=MIDDLE  ALT=\"3%2F2-sqrt%2813%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E3-4x%5E2%2B2x%2B1%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E3-4x%5E2%2B2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <TABLE  BORDER=1>\n" );
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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" );
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document.write( " </SPAN>\n" );
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