document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=758299>Question 758299</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Here is my question:\r<BR>\n" );
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document.write( "when p(x) is divided by x-1 the remainder is 1, and when divided by (x-2)(x-3) the remainder is 5. find remainder when P(X) is divided by (x-1)(x-2)(x-3).\r<BR>\n" );
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document.write( "here is part of my work:<BR>\n" );
document.write( "let the quotient be Q(x), and let the remainder be ax+b<BR>\n" );
document.write( "P(x)=(x-1)(x-2)(x-3)Q(X)+ax+b<BR>\n" );
document.write( "P(1)=a*1+b=1 then a+b=1 (1)\r<BR>\n" );
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document.write( "then I don't  know how to do with (x-2)(x-3) and the remainder is 5. Please help!\r<BR>\n" );
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document.write( "thanks<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 #461409 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=461409\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x-2%29%28x-3%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x-2%29%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is cubic (degree=3) polynomial.<BR>\n" );
document.write( "When you divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x-2%29%28x-3%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x-2%29%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > the remainder polynomial could have degree 2, 1, or 0, so you should start with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a remainder.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28x-1%29%28x-2%29%28x-3%29Q%28X%29%2Bax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"P%28x%29=%28x-1%29%28x-2%29%28x-3%29Q%28X%29%2Bax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%29=a%2A1%5E2%2Bb%2A1%2Bc=a%2Bb%2Bc=1 ALIGN=MIDDLE  ALT=\"P%281%29=a%2A1%5E2%2Bb%2A1%2Bc=a%2Bb%2Bc=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%282%29=a%2A2%5E2%2Bb%2A2%2Bc=4a%2B2b%2Bc=5 ALIGN=MIDDLE  ALT=\"P%282%29=a%2A2%5E2%2Bb%2A2%2Bc=4a%2B2b%2Bc=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%283%29=a%2A3%5E2%2Bb%2A3%2Bc=9a%2B3b%2Bc=5 ALIGN=MIDDLE  ALT=\"P%283%29=a%2A3%5E2%2Bb%2A3%2Bc=9a%2B3b%2Bc=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "That gives you the sytem<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc=5%2C9a%2B3b%2Bc=5%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc=5%2C9a%2B3b%2Bc=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc=5%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%2Bb=4 ALIGN=MIDDLE  ALT=\"3a%2Bb=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bb%2Bc=1%2C9a%2B3b%2Bc=5%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bb%2Bc=1%2C9a%2B3b%2Bc=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a%2B2b=4 ALIGN=MIDDLE  ALT=\"8a%2B2b=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc=5%2C9a%2B3b%2Bc=5%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc=5%2C9a%2B3b%2Bc=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bb%2Bc=1%2C3a%2Bb=4%2C8a%2B2b=4%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bb%2Bc=1%2C3a%2Bb=4%2C8a%2B2b=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283a%2Bb=4%2C8a%2B2b=4%29 ALIGN=MIDDLE  ALT=\"system%283a%2Bb=4%2C8a%2B2b=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a%2B2b-6a-2b=4-8%29 ALIGN=MIDDLE  ALT=\"8a%2B2b-6a-2b=4-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=-4%29 ALIGN=MIDDLE  ALT=\"2a=-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=-2%29 ALIGN=MIDDLE  ALT=\"highlight%28a=-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283a%2Bb=4%2Ca=-2%29 ALIGN=MIDDLE  ALT=\"system%283a%2Bb=4%2Ca=-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%2Bb=4 ALIGN=MIDDLE  ALT=\"-6%2Bb=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=10%29 ALIGN=MIDDLE  ALT=\"highlight%28b=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and finally<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bb%2Bc=1%2Ca=-2%2Cb=10%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bb%2Bc=1%2Ca=-2%2Cb=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2B10%2Bc=1 ALIGN=MIDDLE  ALT=\"-2%2B10%2Bc=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Bc=1 ALIGN=MIDDLE  ALT=\"8%2Bc=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=-7%29 ALIGN=MIDDLE  ALT=\"highlight%28c=-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The remainder  when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divided by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x-2%29%28x-3%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x-2%29%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28-2x%5E2%2B10x-7%29 ALIGN=MIDDLE  ALT=\"highlight%28-2x%5E2%2B10x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "There are infinite possibilities for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as there are for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%28x%29 ALIGN=MIDDLE  ALT=\"Q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. simplest <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > corresponds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%28x%29=1 ALIGN=MIDDLE  ALT=\"Q%28x%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In that case,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28x-1%29%28x-2%29%28x-3%29-2x%5E2%2B10x-7=x%5E3-6x%5E2%2B11x-6-2x%5E2%2B10x-7=x%5E3-8x%5E2%2B21x-13 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );