document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204344>Question 1204344</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3-x%5E2%2B3x%2Bb ALIGN=MIDDLE  ALT=\"ax%5E3-x%5E2%2B3x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divided by x  -  2, the remainder is 59. When it is divided by x + 1, the remainder is  -  1. Find the values of a and b. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840524 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=840524\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Remainder Theorem:<BR>\n" );
document.write( "If P(x) is divided over (x-k), then P(k) is the remainder.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f(x) = ax^3 - x^2 + 3x + b<BR>\n" );
document.write( "f(2) = a(2)^3 - (2)^2 + 3(2) + b<BR>\n" );
document.write( "f(2) = 8a + b + 2<BR>\n" );
document.write( "f(2) = 59 because of the remainder theorem<BR>\n" );
document.write( "8a + b + 2 = 59<BR>\n" );
document.write( "b = 59-2-8a<BR>\n" );
document.write( "b = 57-8a\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f(x) = ax^3 - x^2 + 3x + b<BR>\n" );
document.write( "f(-1) = a(-1)^3 - (-1)^2 + 3(-1) + b<BR>\n" );
document.write( "f(-1) = -a + b - 4<BR>\n" );
document.write( "f(-1) = -1 because of the remainder theorem<BR>\n" );
document.write( "-a + b - 4 = -1<BR>\n" );
document.write( "-a + (57-8a) - 4 = -1<BR>\n" );
document.write( "-9a + 53 = -1<BR>\n" );
document.write( "-9a = -1-53<BR>\n" );
document.write( "-9a = -54<BR>\n" );
document.write( "a = -54/(-9)<BR>\n" );
document.write( "<font color=red size=4>a = 6</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then,<BR>\n" );
document.write( "b = 57-8a<BR>\n" );
document.write( "b = 57 - 8*6<BR>\n" );
document.write( "b = 57 - 48<BR>\n" );
document.write( "<font color=red size=4>b = 9</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );