document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=977664>Question 977664</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all real zeros of the function y=x^3 - 2 x^2 - 25 x +50.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #599227 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52787)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=599227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<B>Solution 1</B>\r<BR>\n" );
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document.write( "The polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+2x%5E2+-+25x+%2B50 ALIGN=MIDDLE  ALT=\"x%5E3+-+2x%5E2+-+25x+%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; has the real root &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Indeed, &nbsp;it is easy to check: &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E3+-2%2A2%5E2+-+25%2A2+%2B+50 ALIGN=MIDDLE  ALT=\"2%5E3+-2%2A2%5E2+-+25%2A2+%2B+50\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Hence, &nbsp;this polynomial can be factored \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+2x%5E2+-+25+x+%2B50 ALIGN=MIDDLE  ALT=\"x%5E3+-+2x%5E2+-+25+x+%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29 ALIGN=MIDDLE  ALT=\"%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "with he linear factor &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2 ALIGN=MIDDLE  ALT=\"x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;where &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; is a quadratic polynomial. \r<BR>\n" );
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document.write( "To find &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;divide the polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+2x%5E2+-+25x+%2B50 ALIGN=MIDDLE  ALT=\"x%5E3+-+2x%5E2+-+25x+%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; by &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2 ALIGN=MIDDLE  ALT=\"x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;(long division). \r<BR>\n" );
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document.write( "If you perform this division, &nbsp;you will get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+2x%5E2+-+25x+%2B50 ALIGN=MIDDLE  ALT=\"x%5E3+-+2x%5E2+-+25x+%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29 ALIGN=MIDDLE  ALT=\"%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2+-+25%29 ALIGN=MIDDLE  ALT=\"%28x%5E2+-+25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Hence, &nbsp;two other roots of the given polynomial are the roots of the quadratic polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+25 ALIGN=MIDDLE  ALT=\"x%5E2+-+25\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;i.e. &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; and &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D ALIGN=MIDDLE  ALT=\"x%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<B>Answer</B>. &nbsp;The roots of the given polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+2x%5E2+-+25x+%2B50 ALIGN=MIDDLE  ALT=\"x%5E3+-+2x%5E2+-+25x+%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; are &nbsp;2, &nbsp;5&nbsp; and &nbsp;-5.\r<BR>\n" );
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document.write( "<B>Solution 2</B>\r<BR>\n" );
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document.write( "You can get factoring the polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+2+x%5E2+-+25+x+%2B50 ALIGN=MIDDLE  ALT=\"x%5E3+-+2+x%5E2+-+25+x+%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; by grouping its terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+2x%5E2+-+25x+%2B50 ALIGN=MIDDLE  ALT=\"x%5E3+-+2x%5E2+-+25x+%2B50\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3+-+2x%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E3+-+2x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825x+-50%29 ALIGN=MIDDLE  ALT=\"%2825x+-50%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%28x-2%29 ALIGN=MIDDLE  ALT=\"x%5E2%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x-2%29 ALIGN=MIDDLE  ALT=\"25%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-25%29%2A%28x-2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2-25%29%2A%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "It gives you the same roots. \r<BR>\n" );
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document.write( " </SPAN>\n" );
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