document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=3549>Question 3549</A>: <I><SPAN STYLE=\"word-break: break-all;\">       factorize\r<BR>\n" );
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document.write( "1.     a(b^4-c^4)+b(c^4-a^4)+c(a^4-b^4) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #1578 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=1578\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">   a(b^4-c^4)+b(c^4-a^4)+c(a^4-b^4)<BR>\n" );
document.write( " = ca^4 -a^4b + ab^4 -b^4c + bc^4 - c^4a<BR>\n" );
document.write( " = ca^4 -b^4c + ab^4 -a^4b - c^4(a-b)<BR>\n" );
document.write( " = c(a^4-b^4) -ab(a^3-b^3) - c^4(a-b)<BR>\n" );
document.write( " = c(a-b)(a+b)(a^2+b^2) -ab(a-b)(a^2+ab+b^2) - c^4(a-b)<BR>\n" );
document.write( " = (a-b)[c(a+b)(a^2+b^2) -ab(a^2+ab+b^2) -c^4]<BR>\n" );
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document.write( " Next, consider<BR>\n" );
document.write( " c(a+b)(a^2+b^2) -ab(a^2+ab+b^2) -c^4<BR>\n" );
document.write( " = c(a+b)(a^2+b^2) -ab(a^2+b^2) -a^2b^2 -c^4<BR>\n" );
document.write( " = (a^2+b^2)[c(a+b) -ab] -a^2b^2 -c^4<BR>\n" );
document.write( " = (a^2+b^2)(c(a+b)-b(a+b))+(a^2+b^2)b^2 -a^2b^2-c^4<BR>\n" );
document.write( " = (a^2+b^2)(a+b)(c-b) + b^4-c^4<BR>\n" );
document.write( " = (b^2+c^2)(b-c)(b+c) -(b-c)(a^2+b^2)(a+b)<BR>\n" );
document.write( " = (b-c)[(b^2+c^2)(b+c) - (a^2+b^2)(a+b)]<BR>\n" );
document.write( " = (b-c)[c^3-a^3 + bc^2 + b^2c - ab^2 - a^2b]<BR>\n" );
document.write( " = (b-c)[(c-a)(c^2+ca+a^2)+ b^2(c - a) + b(c-a)(c+a)]<BR>\n" );
document.write( " = (b-c)(c-a)[(c^2+ca+a^2)+ b^2+ b(c+a)]<BR>\n" );
document.write( " = (b-c)(c-a)[a^2+ b^2+c^2+ab+bc+ca]<BR>\n" );
document.write( " <BR>\n" );
document.write( " Hence, a(b^4-c^4)+b(c^4-a^4)+c(a^4-b^4)<BR>\n" );
document.write( " = (a-b)(b-c)(c-a)[a^2+ b^2+c^2+ab+bc+ca]\r<BR>\n" );
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document.write( " Another way,let f(a,b,c) = a(b^4-c^4)+b(c^4-a^4)+c(a^4-b^4)<BR>\n" );
document.write( " f(a,a,c) = a(a^4-c^4)+a(c^4-a^4) = 0.<BR>\n" );
document.write( " So, (a-b) is a divisor of f(a,b,c).<BR>\n" );
document.write( " Similarly, f(a,b,b) = f(c,b,c) = 0 implies<BR>\n" );
document.write( " (b-c) and (c-a) are factors of f(a,b,c).<BR>\n" );
document.write( " Hence, (a-b)(b-c)(c-a) are factors of f(a,b,c)\r<BR>\n" );
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document.write( " Since degree of f(a,b,c) is 5 and deg of (a-b)(b-c)(c-a) is 3.<BR>\n" );
document.write( " Degree of f(a,b,c)/ (a-b)(b-c)(c-a) should be 2.<BR>\n" );
document.write( " Assume f(a,b,c)= k(a-b)(b-c)(c-a)(a^2+b^2+c^2- h(ab+bc+ca)) <BR>\n" );
document.write( " for some constant k and h [why?]<BR>\n" );
document.write( " Set a=0,b=1,c=-1<BR>\n" );
document.write( " left=1 -(0-1) = 2<BR>\n" );
document.write( " right = k(-1)(2)(-1)(2-h(-1)) = 2k(2+h) = 2<BR>\n" );
document.write( "  k(2+h)=1<BR>\n" );
document.write( " Set a=0,b=1,c=2<BR>\n" );
document.write( " left hand side = 16 + 2(-1) = 14<BR>\n" );
document.write( " right hand side = k(-1)(-1)(2)(5-h(2)) = 2k(5-2h) = 14<BR>\n" );
document.write( " k(5-2h) = 7<BR>\n" );
document.write( " 5-2h = 7, h =  -1, k = 1<BR>\n" );
document.write( " So, a(b^4-c^4)+b(c^4-a^4)+c(a^4-b^4)<BR>\n" );
document.write( " = (a-b)(b-c)(c-a)(a^2+b^2+c^2+ ab+bc+ca)<BR>\n" );
document.write( " (same answer)\r<BR>\n" );
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document.write( " Kenny </SPAN>\n" );
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