document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023621>Question 1023621</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help and show all work. Thanks\r<BR>\n" );
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document.write( " Real Zeros of Polynomials\r<BR>\n" );
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document.write( "1. The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=3 and roots of multiplicity 1 at x=0 and x=&#8722;2. It goes through the point (5,14). \r<BR>\n" );
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document.write( "Find a formula for P(x). \r<BR>\n" );
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document.write( "P(x)=\r<BR>\n" );
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document.write( "2. The polynomial of degree 3, P(x), has a root of multiplicity 2 at x=5 and root of multiplicity 1 at x=-1. The y-intercept is y=-12.5. \r<BR>\n" );
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document.write( "Find a formula for P(x). \r<BR>\n" );
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document.write( "P(x)= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #639175 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=639175\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. A polynomial with a root of multiplicity 2 at x=3 has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a factor.<BR>\n" );
document.write( "A polynomial with roots of multiplicity 1 at x=0 and x=&#8722;2<BR>\n" );
document.write( "has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-0%29=x ALIGN=MIDDLE  ALT=\"%28x-0%29=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-%28-2%29%29=%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x-%28-2%29%29=%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as factors.<BR>\n" );
document.write( "The factors listed make<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=a%2Ax%2A%28x%2B2%29%2A%28x-3%29%5E2 ALIGN=MIDDLE  ALT=\"P%28x%29=a%2Ax%2A%28x%2B2%29%2A%28x-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E0 ALIGN=MIDDLE  ALT=\"a%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > a polynomial f degree 4.<BR>\n" );
document.write( "If the graph of such polynomial goes through point (5,14), then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%285%29=14 ALIGN=MIDDLE  ALT=\"P%285%29=14\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A5%2A%285%2B2%29%2A%285-3%29%5E2=14 ALIGN=MIDDLE  ALT=\"a%2A5%2A%285%2B2%29%2A%285-3%29%5E2=14\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A5%2A7%2A2%5E2=14 ALIGN=MIDDLE  ALT=\"a%2A5%2A7%2A2%5E2=14\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A5%2A7%2A4=14 ALIGN=MIDDLE  ALT=\"a%2A5%2A7%2A4=14\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=14%2F%285%2A7%2A4%29 ALIGN=MIDDLE  ALT=\"a=14%2F%285%2A7%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0.1 ALIGN=MIDDLE  ALT=\"a=0.1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28P%28x%29=0.1%2Ax%2A%28x%2B2%29%2A%28x-3%29%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28P%28x%29=0.1%2Ax%2A%28x%2B2%29%2A%28x-3%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a formula for such a polynomial.<BR>\n" );
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document.write( "2. Similarly, the polynomial must be of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=a%2A%28x%2B1%29%2A%28x-5%29%5E2 ALIGN=MIDDLE  ALT=\"P%28x%29=a%2A%28x%2B1%29%2A%28x-5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E0 ALIGN=MIDDLE  ALT=\"a%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "ana if the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12.5 ALIGN=MIDDLE  ALT=\"-12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=12.5 ALIGN=MIDDLE  ALT=\"P%280%29=12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A%280%2B1%29%2A%280-5%29%5E2=12.5 ALIGN=MIDDLE  ALT=\"a%2A%280%2B1%29%2A%280-5%29%5E2=12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A1%2A25=12.5 ALIGN=MIDDLE  ALT=\"a%2A1%2A25=12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A25=12.5 ALIGN=MIDDLE  ALT=\"a%2A25=12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=12.5%2F25 ALIGN=MIDDLE  ALT=\"a=12.5%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0.5 ALIGN=MIDDLE  ALT=\"a=0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28P%28x%29=0.5%2A%28x%2B1%29%2A%28x-5%29%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28P%28x%29=0.5%2A%28x%2B1%29%2A%28x-5%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a formula for such a polynomial. </SPAN>\n" );
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