document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=995119>Question 995119</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I really need help with this problem. Please show every step by step. Thank you\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The polynomial of degree 4, P(x)  has a root of multiplicity 2 at x=2  and roots of multiplicity 1 at x=0  and x=&#8722;2  . It goes through the point (5,252)  . <BR>\n" );
document.write( "Find a formula for 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 #613950 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52903)</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=613950\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "The polynomial of degree 4, P(x)  has a root of multiplicity 2 at x=2  and roots of multiplicity 1 at x=0  and x=&#8722;2. It goes through the point (5,252). <BR>\n" );
document.write( "Find a formula for P(x).<BR>\n" );
document.write( "----------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "According to the given information about the roots, &nbsp;the polynomial is \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "P(x) = &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A%28x-2%29%5E2%2A%28x-0%29%2A%28x-%28-2%29%29 ALIGN=MIDDLE  ALT=\"a%2A%28x-2%29%5E2%2A%28x-0%29%2A%28x-%28-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A%28x-2%29%5E2%2Ax%2A%28x%2B2%29 ALIGN=MIDDLE  ALT=\"a%2A%28x-2%29%5E2%2Ax%2A%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "with some unknown real coefficient &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To find &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;use the fact that the plot of the polynomial goes through the point &nbsp;(5,252). &nbsp;It means that &nbsp;P(5) = 252. \r<BR>\n" );
document.write( "\n" );
document.write( "So, &nbsp;simply substitute &nbsp;x=5 &nbsp;into the polynomial &nbsp;(1). You will get the equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A%285-2%29%5E2%2A5%2A%285%2B2%29 ALIGN=MIDDLE  ALT=\"a%2A%285-2%29%5E2%2A5%2A%285%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=252 ALIGN=MIDDLE  ALT=\"252\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;or\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A3%5E2%2A5%2A7 ALIGN=MIDDLE  ALT=\"a%2A3%5E2%2A5%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=252 ALIGN=MIDDLE  ALT=\"252\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;or\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=315a ALIGN=MIDDLE  ALT=\"315a\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=252 ALIGN=MIDDLE  ALT=\"252\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hence, &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=252%2F315 ALIGN=MIDDLE  ALT=\"252%2F315\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F5 ALIGN=MIDDLE  ALT=\"4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, &nbsp;P(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F5 ALIGN=MIDDLE  ALT=\"4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2Ax%2A%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2Ax%2A%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You can open the parentheses if you need.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );