document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=204562>Question 204562</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi there, I have an exam today and I was doing a practise exam where I came across this Polynomial and rational functions question and was unable to solve it. Could one of the tutors please help me with it, I'd be more than grateful!!!!\r<BR>\n" );
document.write( "\n" );
document.write( "Find the remainder:(x^4 - 2x^2 + 5 - 2)/(x - 3). (2 marks) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156323 by </B><A HREF=/tutors/aboutme.mpl?userid=mickclns><B>mickclns(59)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mickclns><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=156323\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Did you forget the x after the 5?  I am assuming you did, so we have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E4+-+2x%5E2+%2B+5x+-+2%29%2F%28x+-+3%29 ALIGN=MIDDLE  ALT=\"%28x%5E4+-+2x%5E2+%2B+5x+-+2%29%2F%28x+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "You could use long (polynomial) division, but since the divisor is linear, synthetic division is easier:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C150%2C0%2C100%2C50%2C100%2C%0D%0A+++++locate%2820%2C90%2C3%29%2C%0D%0A%0D%0A+++++locate%2830%2C90%2C1%29%2C%0D%0A+++++locate%2840%2C90%2C0%29%2C%0D%0A+++++locate%2848%2C90%2C-2%29%2C%0D%0A+++++locate%2860%2C90%2C5%29%2C%0D%0A+++++locate%2869%2C90%2C-2%29%2C%0D%0A%0D%0A+++++locate%2840%2C80%2C3%29%2C%0D%0A+++++locate%2850%2C80%2C9%29%2C%0D%0A+++++locate%2859%2C80%2C21%29%2C%0D%0A+++++locate%2869%2C80%2C78%29%2C%0D%0A%0D%0A+++++locate%2830%2C70%2C1%29%2C%0D%0A+++++locate%2840%2C70%2C3%29%2C%0D%0A+++++locate%2850%2C70%2C7%29%2C%0D%0A+++++locate%2859%2C70%2C26%29%2C%0D%0A+++++locate%2869%2C70%2C76%29%2C%0D%0A%0D%0A+++++line%2825%2C90%2C25%2C72%29%2C%0D%0A+++++line%2875%2C72%2C25%2C72%29%2C%0D%0A%0D%0A+++++line%2867%2C72%2C67%2C64%29%2C%0D%0A+++++line%2867%2C64%2C75%2C64%29%0D%0A%0D%0A%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The 3 at upper left is the root of x-3, the linear divisor.<BR>\n" );
document.write( "The 1 to its right is the coefficient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4 ALIGN=MIDDLE  ALT=\"x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "As we move to the right we have the rest of the coefficients, 0, -2, 5, -2, including the (necessary) 0 for the non-existent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > term.\r<BR>\n" );
document.write( "\n" );
document.write( "The first coefficient, 1 in this case, we always bring straight down and put below the (long) horizontal line.  Then we multiply that 1 by the 3 and put the resulting 3 in the second row, second column. Next, we add THAT 3 to the 0 above it and put the resulting 3 below the line.  That 3 gets multiplied by the original 3 (root of x-3) and the resulting 9 in the second row, third column.  <BR>\n" );
document.write( "Then repeat: 9 + -2 is 7, 7 x 3 is 21, 21 + 5 is 26, 26 x 3 is 78, 78 + -2 is 76, which is where we stop because we have no more columns.  The bottom right number is our remainder and is separated from the others by lines for that reason.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Three things to notice about this result:<BR>\n" );
document.write( "1) if it is 0, then the denominator divides the numerator in the original fraction evenly.<BR>\n" );
document.write( "2) the remainder is also what you get when you plug the top right number into the numerator.  In other words, in this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283%29%5E4+-+2%283%29%5E2+%2B+5%283%29+-+2%29+=+76 ALIGN=MIDDLE  ALT=\"%28%283%29%5E4+-+2%283%29%5E2+%2B+5%283%29+-+2%29+=+76\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  This is usually a much easier process than taking 3 to the 4th power, subtracting from it ... . When used in this way it is called synthetic substitution.<BR>\n" );
document.write( "3) Look back at the diagram. This shows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E4+-+2x%5E2+%2B+5x+-+2%29%2F%28x+-+3%29+=+x%5E3+%2B+3x%5E2+%2B+7x+%2B+26+%2B+76%2F%28x-3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, which is also what would have resulted from using long division. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );