document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=814185>Question 814185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a polynomial satisfying the following conditions:\r<BR>\n" );
document.write( "\n" );
document.write( "f has degree 3;the roots of f are precisely 2,3,4;the leading coefficient is 2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #490192 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=490192\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If \"r\" is a root of a polynomial, then (x - r) is a factor of the polynomial. So with the three given roots and the leading coefficient we can immediately write a factored version of the desired function:<BR>\n" );
document.write( "f(x) = 2(x-2)(x-3)(x-4)<br><BR>\n" );
document.write( "All that is left is to multiply this out. Since this is all multiplication we can multiply in any order. I'll start by using FOIL on the last two factors:<BR>\n" );
document.write( "f(x) = 2(x-2)(x*x+x*(-4)+(-3)*x+(-3)(-4))<BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2%28x-2%29%28x%5E2%2B%28-4%29x%2B%28-3%29x%2B12%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2%28x-2%29%28x%5E2%2B%28-4%29x%2B%28-3%29x%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2%28x-2%29%28x%5E2%2B%28-7%29x%2B12%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2%28x-2%29%28x%5E2%2B%28-7%29x%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying the first two factors:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%282x-4%29%28x%5E2%2B%28-7%29x%2B12%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%282x-4%29%28x%5E2%2B%28-7%29x%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "One more multiplication:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%2Ax%5E2%2B2x%2A%28-7%29x%2B2x%2A12%2B%28-4%29%2Ax%5E2%2B%28-4%29%2A%28-7%29x%2B%28-4%29%2A12%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%5E3%2B%28-14%29x%5E2%2B24x%2B%28-4%29%2Ax%5E2%2B28x%2B%28-48%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2x%5E3%2B%28-14%29x%5E2%2B24x%2B%28-4%29%2Ax%5E2%2B28x%2B%28-48%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%5E3%2B%28-18%29x%5E2%2B52x%2B%28-48%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2x%5E3%2B%28-18%29x%5E2%2B52x%2B%28-48%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%5E3-18x%5E2%2B52x-48 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2x%5E3-18x%5E2%2B52x-48\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );