document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1077691>Question 1077691</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If f is a polynomial of degree 4 such that<BR>\n" );
document.write( "f(0) = f(1) = f(2) = f(3) = 1<BR>\n" );
document.write( "and<BR>\n" );
document.write( "f(4) = 0,<BR>\n" );
document.write( "then determine f(5). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #692173 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=692173\">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=f%28x%29+=+ax%5E4+%2B+bx%5E3+%2B+cx%5E2+%2B+dx+%2B+e+ ALIGN=MIDDLE  ALT=\"f%28x%29+=+ax%5E4+%2B+bx%5E3+%2B+cx%5E2+%2B+dx+%2B+e+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%280%29+=+1+=+0%2B0%2B0%2B0%2Be+ ALIGN=MIDDLE  ALT=\"+f%280%29+=+1+=+0%2B0%2B0%2B0%2Be+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   —> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e+=+1+ ALIGN=MIDDLE  ALT=\"+e+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (use e=1 in evaluating f(1)…f(4) below)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%281%29+=+1+ ALIGN=MIDDLE  ALT=\"+f%281%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > —> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+%2B+b+%2B+c+%2B+d+%2B+e+=+1+ ALIGN=MIDDLE  ALT=\"+a+%2B+b+%2B+c+%2B+d+%2B+e+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > —> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+%2B+c+%2B+d+=+0+ ALIGN=MIDDLE  ALT=\"a+%2B+b+%2B+c+%2B+d+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%282%29+=+1+ ALIGN=MIDDLE  ALT=\"+f%282%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > —> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%2816%29+%2B+b%288%29+%2B+c%284%29+%2B+d%282%29+%2B+e+=+1+ ALIGN=MIDDLE  ALT=\"+a%2816%29+%2B+b%288%29+%2B+c%284%29+%2B+d%282%29+%2B+e+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > —> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+16a+%2B+8b+%2B+4c+%2B+2d+=+0+ ALIGN=MIDDLE  ALT=\"+16a+%2B+8b+%2B+4c+%2B+2d+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%283%29+=+1+ ALIGN=MIDDLE  ALT=\"+f%283%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > —> ... —> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+81a+%2B+27b+%2B+9c+%2B+d+=+0+ ALIGN=MIDDLE  ALT=\"+81a+%2B+27b+%2B+9c+%2B+d+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%284%29+=+0+ ALIGN=MIDDLE  ALT=\"+f%284%29+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > —> … —> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+256a+%2B+64b+%2B+16c+%2B+4d+%2B+1+=+0+ ALIGN=MIDDLE  ALT=\"+256a+%2B+64b+%2B+16c+%2B+4d+%2B+1+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "—<BR>\n" );
document.write( "This generates the following matrix (moving constants to right hand side):<BR>\n" );
document.write( "—<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28matrix%284%2C4%2C+1%2C1%2C1%2C1%2C+16%2C8%2C4%2C2%2C++81%2C27%2C9%2C3%2C++256%2C64%2C16%2C4%29%29+%2A+%28matrix%284%2C1%2C+a%2Cb%2Cc%2Cd%29%29+=+%28matrix%284%2C1%2C+0%2C0%2C0%2C-1%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "The inverse of the 4x4 matrix is (from https://www.symbolab.com/solver/matrix-inverse-calculator )\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28matrix%284%2C4%2C++-1%2F6%2C+1%2F4%2C+-1%2F6%2C+1%2F24%2C+++3%2F2%2C-2%2C+7%2F6%2C+-1%2F4%2C++-13%2F3%2C+19%2F4%2C+-7%2F3%2C+11%2F24%2C+++4%2C-3%2C4%2F3%2C-1%2F4%29%29++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "--<BR>\n" );
document.write( "And the solution we seek is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28matrix%284%2C1%2Ca%2Cb%2Cc%2Cd%29%29+=+%28matrix%284%2C4%2C++-1%2F6%2C+1%2F4%2C+-1%2F6%2C+1%2F24%2C+++3%2F2%2C-2%2C+7%2F6%2C+-1%2F4%2C++-13%2F3%2C+19%2F4%2C+-7%2F3%2C+11%2F24%2C+++4%2C-3%2C4%2F3%2C-1%2F4%29%29%2A%28matrix%284%2C1%2C+0%2C0%2C0%2C-1%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+-1%2F6+%2B+1%2F4+-+1%2F6+-+1%2F24+=+-3%2F24+ ALIGN=MIDDLE  ALT=\"+a+=+-1%2F6+%2B+1%2F4+-+1%2F6+-+1%2F24+=+-3%2F24+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+3%2F2+-+2+%2B+7%2F6+%2B+1%2F4+=+22%2F24+ ALIGN=MIDDLE  ALT=\"+b+=+3%2F2+-+2+%2B+7%2F6+%2B+1%2F4+=+22%2F24+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+-57%2F24+ ALIGN=MIDDLE  ALT=\"+c+=+-57%2F24+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d+=+62%2F24+ ALIGN=MIDDLE  ALT=\"+d+=+62%2F24+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28-3%2F24%29x%5E4+%2B+%2822%2F24%29x%5E3+%2B+%28-57%2F24%29x%5E2+%2B+%2862%2F24%29x+%2B+1+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "pull out 1/24:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%281%2F24%29%28-3x%5E4+%2B+22x%5E3+-57x%5E2+%2B+62x+%2B+24%29+ ALIGN=MIDDLE  ALT=\"f%28x%29+=+%281%2F24%29%28-3x%5E4+%2B+22x%5E3+-57x%5E2+%2B+62x+%2B+24%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "—<BR>\n" );
document.write( "Now we are ready to find f(5):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%285%29+=+%281%2F24%29%28-3%28625%29+%2B+22%28125%29+-57%2825%29+%2B+62%285%29+%2B+24%29+ ALIGN=MIDDLE  ALT=\"+f%285%29+=+%281%2F24%29%28-3%28625%29+%2B+22%28125%29+-57%2825%29+%2B+62%285%29+%2B+24%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%285%29+=+%281%2F24%29%28-216%29+=+highlight%28-9%29+ ALIGN=MIDDLE  ALT=\"+f%285%29+=+%281%2F24%29%28-216%29+=+highlight%28-9%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );