document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=927008>Question 927008</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let P(x) = kx^3 + 2k^2x^2 + k^3. Find the sum of all real numbers k for which x-2 is a factor of P(x).\r<BR>\n" );
document.write( "\n" );
document.write( "Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #562630 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=562630\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Using the remainder theorem, if x-k is a factor of P(x), then P(k) = 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So that means if x-2 is a factor of P(x), then P(2) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug x = 2 into P(x) and set it equal to 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "P(x) = kx^3 + 2k^2x^2 + k^3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "P(x) = k(2)^3 + 2k^2(2)^2 + k^3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "P(x) = 8k+8k^2+k^3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "0 = 8k+8k^2+k^3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "8k+8k^2+k^3 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "k^3+8k^2+8k = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now we solve for k\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "k^3+8k^2+8k = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "k*(k^2+8k+8) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I'll let you finish up. One of those factors will give you complex (nonreal) solutions.<BR>\n" );
document.write( "------------------------------------------------------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you need more one-on-one help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help\">jim_thompson5910@hotmail.com</a>. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Alternatively, please consider visiting my website: <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">http://www.freewebs.com/jimthompson5910/home.html</a> and making a donation. Any amount is greatly appreciated as it helps me a lot. This donation is to support free tutoring. Thank you.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Jim\r<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------------------------------------------------------------------ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );