document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170553>Question 1170553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The expression ax(squared)+bx+c is divisible by x-1 and has a remainder of 2 when divided by x+1 and has a remainder of 8 when divided by x-2. Find the value of a, b and c </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #795409 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=795409\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x^2 = x squared\r<BR>\n" );
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document.write( "p(x) = ax^2 + bx + c\r<BR>\n" );
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document.write( "When we divide p(x) over (x-k), the remainder is p(k) according to the remainder theorem.\r<BR>\n" );
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document.write( "p(x) is divisible by (x-1), so the remainder is 0<BR>\n" );
document.write( "So p(1) = 0<BR>\n" );
document.write( "This means p(x) = (x-1)*q(x) for some polynomial q(x) \r<BR>\n" );
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document.write( "When we divide p(x) over (x+1), the remainder is 2<BR>\n" );
document.write( "So p(-1) = 2, since k = -1 here.\r<BR>\n" );
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document.write( "When we divide p(x) over (x-2), the remainder is 8. <BR>\n" );
document.write( "So p(2) = 8\r<BR>\n" );
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document.write( "We have these function statements<BR>\n" );
document.write( "p(1) = 0<BR>\n" );
document.write( "p(-1) = 2<BR>\n" );
document.write( "p(2) = 8\r<BR>\n" );
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document.write( "This is another way of saying we know the three points <BR>\n" );
document.write( "(1,0)<BR>\n" );
document.write( "(-1,2)<BR>\n" );
document.write( "(2,8)<BR>\n" );
document.write( "are on the p(x) curve.\r<BR>\n" );
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document.write( "Plug in (x,y) = (1,0) into p(x) to get<BR>\n" );
document.write( "p(x) = ax^2 + bx + c<BR>\n" );
document.write( "p(1) = a(1)^2 + b(1) + c<BR>\n" );
document.write( "0 = a(1)^2 + b(1) + c<BR>\n" );
document.write( "a+b+c = 0<BR>\n" );
document.write( "Solve for c<BR>\n" );
document.write( "c = -a-b\r<BR>\n" );
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document.write( "Plug (-1,2) into p(x)<BR>\n" );
document.write( "p(x) = ax^2 + bx + c<BR>\n" );
document.write( "p(-1) = a(-1)^2 + b(-1) + c<BR>\n" );
document.write( "2 = a - b + c<BR>\n" );
document.write( "a - b + c = 2<BR>\n" );
document.write( "Plug in c = -a-b<BR>\n" );
document.write( "a - b + c = 2<BR>\n" );
document.write( "a - b + (-a-b) = 2<BR>\n" );
document.write( "-2b = 2<BR>\n" );
document.write( "b = -1\r<BR>\n" );
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document.write( "Use that b value to get<BR>\n" );
document.write( "c = -a-b<BR>\n" );
document.write( "c = -a-(-1)<BR>\n" );
document.write( "c = -a+1\r<BR>\n" );
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document.write( "Now plug (x,y) = (2,8) into p(x)<BR>\n" );
document.write( "p(x) = ax^2 + bx + c<BR>\n" );
document.write( "p(2) = a(2)^2 + b(2) + c<BR>\n" );
document.write( "p(2) = 4a + 2b + c<BR>\n" );
document.write( "8 = 4a + 2b + c<BR>\n" );
document.write( "4a + 2b + c = 8\r<BR>\n" );
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document.write( "Plug in c = -a+1 and b = -1. Solve for 'a'<BR>\n" );
document.write( "4a + 2b + c = 8<BR>\n" );
document.write( "4a + 2(-1) + (-a+1) = 8<BR>\n" );
document.write( "4a - 2 - a + 1 = 8<BR>\n" );
document.write( "3a - 1 = 8<BR>\n" );
document.write( "3a = 8+1<BR>\n" );
document.write( "3a = 9<BR>\n" );
document.write( "a = 9/3<BR>\n" );
document.write( "a = 3\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "c = -a+1<BR>\n" );
document.write( "c = -3+1<BR>\n" );
document.write( "c = -2\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "a = 3 <BR>\n" );
document.write( "b = -1<BR>\n" );
document.write( "c = -2<BR>\n" );
document.write( "The polynomial is p(x) = 3x^2-x-2\r<BR>\n" );
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