document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=885552>Question 885552</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What are the zeroes of f(x) = x^3 + 5x^2 - 7x + 1? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #535227 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=535227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = x^3 + 5x^2 - 7x + 1<BR>\n" );
document.write( "possible zeros are -1 and 1.<BR>\n" );
document.write( "use synthetic division to check<BR>\n" );
document.write( "1 | 1  5  -7  1<BR>\n" );
document.write( "       1   6  -1<BR>\n" );
document.write( "   -----------------<BR>\n" );
document.write( "    1  6  -1   0<BR>\n" );
document.write( "since the remainder is zero it is a factor:<BR>\n" );
document.write( "(x-1)(x^2+6x-1)<BR>\n" );
document.write( "normally, you would factor the quadratic but since it can't be factored we use the \"quadratic formula\" to get<BR>\n" );
document.write( "zeroes are at:<BR>\n" );
document.write( "x = 1; x = 0.1623; and x = -6.1623<BR>\n" );
document.write( ".<BR>\n" );
document.write( "details of quadratic equation follows:<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=quadratic.solver>SOLVE quadratic equation with variable</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%2B6x%2B-1+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B6x%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=%286%29%5E2-4%2A1%2A-1=40 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%286%29%5E2-4%2A1%2A-1=40\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=40 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+=+%28-6%2B-sqrt%28+40+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-6%2B-sqrt%28+40+%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-%286%29%2Bsqrt%28+40+%29%29%2F2%5C1+=+0.16227766016838 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%286%29%2Bsqrt%28+40+%29%29%2F2%5C1+=+0.16227766016838\"   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-%286%29-sqrt%28+40+%29%29%2F2%5C1+=+-6.16227766016838 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%286%29-sqrt%28+40+%29%29%2F2%5C1+=+-6.16227766016838\"   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%2B6x%2B-1 ALIGN=MIDDLE  ALT=\"1x%5E2%2B6x%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%2B6x%2B-1+=+1%28x-0.16227766016838%29%2A%28x--6.16227766016838%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B6x%2B-1+=+1%28x-0.16227766016838%29%2A%28x--6.16227766016838%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.16227766016838, -6.16227766016838.\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%2B6%2Ax%2B-1+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B6%2Ax%2B-1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );