document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=41563>Question 41563</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the equality x^3 = A(x - 1)(x - 2)(x - 3) + B(x - 1)(x - 2) + C(x - 2) + D is an identity, what is A + B + C + D? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #26838 by </B><A HREF=/tutors/aboutme.mpl?userid=kev82><B>kev82(151)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kev82><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=26838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
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document.write( "Because it's an identity you can shove in any values you like for x and get some simultaneous equations to solve. However solving a 4x4 set of simultaneous equations isn't much fun. By factoring we can choose suitable values of x that let us read off the values of the coefficients.\r<BR>\n" );
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document.write( "Factoring gives\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x^3=D+(x-2)(C+(x-1)(B+A(x-3)))\">\r<BR>\n" );
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document.write( "Taking x=2 gives D=8\r<BR>\n" );
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document.write( "Then rearranging to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\frac{x^3-8}{x-2}=C+(x-1)(B+A(x-3))\">\r<BR>\n" );
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document.write( "And this time choosing x=1 gives C=7\r<BR>\n" );
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document.write( "Rearranging again (it looks messy but it's OK, trust me!)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\frac{\frac{x^3-8}{x-2}-7}{x-1}=B+A(x-3)\">\r<BR>\n" );
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document.write( "Now choosing x=3 gives B=6\r<BR>\n" );
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document.write( "Finally\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\frac{\frac{x^3-8}{x-2}-7}{x-1}-6=A(x-3)\">\r<BR>\n" );
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document.write( "Choose x=4 to find A=1\r<BR>\n" );
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document.write( "8+7+6+1=22, which is your answer.\r<BR>\n" );
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document.write( "Hope that helps,<BR>\n" );
document.write( "Kev\r<BR>\n" );
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document.write( "<i>Message from question poster</i><BR>\n" );
document.write( "<b>I accidentally messed up the end of the problem, it should have been <BR>\n" );
document.write( "C(x - 1), but using your problem I was able to come up with the solution. I had decided while working the problem that I would have to factor in some way, but I had not thought of factoring in that manner. I also knew the fact that it was an identity had to come into play. <BR>\n" );
document.write( "Fixing the problem I got A=1, B = 6, C = 7 and D=1.<BR>\n" );
document.write( "Can you tell me if this is correct??</b>\r<BR>\n" );
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document.write( "Yes, it is. You could of course check the answer by multiplying out all the brackets and you should find you getsomething which is always true. Like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=0 ALIGN=MIDDLE  ALT=\"0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
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