document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1080238>Question 1080238</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the zeros of the polynomial function and state the multiplicity of each <BR>\n" );
document.write( "F(x) = 5(x+7)^2 (x-7)^3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #694424 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=694424\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: the term \"zeros\" is the same as \"roots\"\r<BR>\n" );
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document.write( "We have this as our original function<BR>\n" );
document.write( "F(x) = 5(x+7)^2 (x-7)^3\r<BR>\n" );
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document.write( "I'm going to highlight the exponents in different colors <font color=red>red</font> and <font color=blue>blue</font> so things are separated clearly.<BR>\n" );
document.write( "F(x) = 5(x+7)^<font color=red>2</font> (x-7)^<font color=blue>3</font><BR>\n" );
document.write( "These exponents will be important for identifying the multiplicity.\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "To find the roots, we need to set f(x) equal to zero and solve for x\r<BR>\n" );
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document.write( "F(x) = 0<BR>\n" );
document.write( "5(x+7)^2 (x-7)^3 = 0<BR>\n" );
document.write( "(x+7)^2 = 0 or (x-7)^3 = 0<BR>\n" );
document.write( "x+7= 0 or x-7 = 0<BR>\n" );
document.write( "x = -7 or x = 7\r<BR>\n" );
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document.write( "So the two distinct roots are x = -7 or x = 7\r<BR>\n" );
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document.write( "The root x = -7 has multiplicity <font color=red>2</font>. Take note of the specific color coding. This <font color=red>2</font>  is an exponent for the term (x+7) which is where the root is derived from.\r<BR>\n" );
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document.write( "Similarly, the root x = 7 has multiplicity <font color=blue>3</font> because the <font color=blue>3</font> is the exponent for (x-7). \r<BR>\n" );
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document.write( "Extra info: an even multiplicity (such as 2) means that the graph touches the x axis and turns around. Think of a parabola like shape. See the graph below where point A is located for a visual example. An odd multiplicity root is one where the graph crosses over the x axis and keeps going (though it might turn around at some point). A visual example of this is the root where point B is located in the graph below.\r<BR>\n" );
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document.write( "Here is what the graph looks like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/YqsWoLX.png\"><BR>\n" );
document.write( "Image generated by <a  rel=nofollow HREF= \"https://www.geogebra.org/home\">GeoGebra</a> (free graphing software).<BR>\n" );
document.write( "The arrows indicate that the graph continues on forever along that general curve path (but the window cuts things off).  </SPAN>\n" );
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