document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=498249>Question 498249</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which statement about the function below is true?<BR>\n" );
document.write( "f(x)+x^3 - 5x^2 - 25x + 125 = (x+5)(x-5)(x-5)\r<BR>\n" );
document.write( "\n" );
document.write( "A. The root -5 has a multiplicity of 1<BR>\n" );
document.write( "B. The root -5 has a multiplicity of 2<BR>\n" );
document.write( "C. The root 5 has a multiplicity of 1<BR>\n" );
document.write( "D. The root 5 has a multiplicity of 3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #336982 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=336982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your factors of the equation of:<BR>\n" );
document.write( "x^3 - 5x^2 - 25x + 125 are:<BR>\n" );
document.write( "(x - 5) * (x - 5) * (x + 5)<BR>\n" );
document.write( "this means that the possible solutions to this equation are:<BR>\n" );
document.write( "x = 5<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x = -5<BR>\n" );
document.write( "x = 5 occurs twice because the (x - 5) factor occurs twice.<BR>\n" );
document.write( "therefore x = 5 has a multiplicity of 2.<BR>\n" );
document.write( "x = -5 occurs once because the (x + 5) factor occurs once.<BR>\n" );
document.write( "$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$<BR>\n" );
document.write( "the answer to your question is:<BR>\n" );
document.write( "A.  The root -5 has a multiplicity of 1.<BR>\n" );
document.write( "$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$<BR>\n" );
document.write( "here's a reference on multiplicity of roots that you might be interested in.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx\" target = \"_blank\">http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx</a><BR>\n" );
document.write( "scroll down to the sentence that says:<BR>\n" );
document.write( "\"Now, we’ve got some terminology to get out of the way.  If r is a zero of a polynomial and the exponent on the term that produced the root is k then we say that r has multiplicity k.  Zeroes with a multiplicity of 1 are often called simple zeroes.\"<BR>\n" );
document.write( "That's where the discussion about multiplicity of roots starts.<BR>\n" );
document.write( "it says what i just told you.<BR>\n" );
document.write( "the multiplicity of the root is the number of times it is a factor of the equation.<BR>\n" );
document.write( "If we graph your equation of:<BR>\n" );
document.write( "x^3 - 5x^2 - 25x + 125, then we will get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C10%2C-200%2C200%2Cx%5E3+-+5x%5E2+-+25x+%2B+125%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C10%2C-200%2C200%2Cx%5E3+-+5x%5E2+-+25x+%2B+125%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you can see that the graph touches the x-axis at x = -5 and x = 5.<BR>\n" );
document.write( "the graph, however, does not show you that the value of x = 5 occurs 2 times in the solution of the equation.<BR>\n" );
document.write( "that comes from the examination of the factors of the equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );