document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=656264>Question 656264</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ``the function f(x)=x(cubed)-2x(squared)-5x+6 has three integer roots<BR>\n" );
document.write( " <BR>\n" );
document.write( "1) Find the tree roots?<BR>\n" );
document.write( " 2)Find a cubic euation wose roots are 1 less than the roots of f.<BR>\n" );
document.write( " <BR>\n" );
document.write( "PLEASE HELP ME''.<BR>\n" );
document.write( "  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #409416 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=409416\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Factors of 6: 1, 2, 3, 6, -1, -2, -3, -6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factors of 1: 1, -1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "*** Note: Include negative factors as well ***\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divide all the factors of 6 by the factors of 1 to get the potential rational roots of x^3-2x^2-5x+6: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1, -1, 2, -2, 3, -3, 6, -6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: there are 8 possible rational roots (listed above) for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=====================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's check all the possible roots to see if there are any actual rational roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check to see if x = 1 is a root for x^3-2x^2-5x+6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(1)^3-2(1)^2-5(1)+6 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the result is 0, x = 1 is a root for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Check to see if x = -1 is a root for x^3-2x^2-5x+6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(-1)^3-2(-1)^2-5(-1)+6 = 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the result is NOT 0, x = -1 is NOT a root for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Check to see if x = 2 is a root for x^3-2x^2-5x+6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(2)^3-2(2)^2-5(2)+6 = -4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the result is NOT 0, x = 2 is NOT a root for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Check to see if x = -2 is a root for x^3-2x^2-5x+6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(-2)^3-2(-2)^2-5(-2)+6 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the result is 0, x = -2 is a root for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Check to see if x = 3 is a root for x^3-2x^2-5x+6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(3)^3-2(3)^2-5(3)+6 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the result is 0, x = 3 is a root for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Check to see if x = -3 is a root for x^3-2x^2-5x+6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(-3)^3-2(-3)^2-5(-3)+6 = -24\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the result is NOT 0, x = -3 is NOT a root for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Check to see if x = 6 is a root for x^3-2x^2-5x+6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(6)^3-2(6)^2-5(6)+6 = 120\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the result is NOT 0, x = 6 is NOT a root for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Check to see if x = -6 is a root for x^3-2x^2-5x+6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(-6)^3-2(-6)^2-5(-6)+6 = -252\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the result is NOT 0, x = -6 is NOT a root for x^3-2x^2-5x+6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=====================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The polynomial x^3-2x^2-5x+6 has 3 rational roots and they are: x = 1, x = -2, x = 3, \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the rational factors are: (x - 1), (x + 2) and (x - 3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means that x^3-2x^2-5x+6 factors to (x - 1)(x + 2)(x - 3) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );