document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=373641>Question 373641</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the cubic polynomial which takes the following values y(0) = 1, y(1) = 0, y(2) = 1 and y(3) = 10 , hence obtain the value of y(0.5)?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #265947 by </B><A HREF=/tutors/aboutme.mpl?userid=CharlesG2><B>CharlesG2(834)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CharlesG2><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=265947\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the cubic polynomial which takes the following values y(0) = 1, y(1) = 0, y(2) = 1 and y(3) = 10 , hence obtain the value of y(0.5)?\r<BR>\n" );
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document.write( "y = ax^3 + bx^2 + cx + d<BR>\n" );
document.write( "y(0) = 1 = 0 + 0 + 0 + d, d = 1<BR>\n" );
document.write( "y(1) = 0 = a + b + c + 1, a + b + c = -1<BR>\n" );
document.write( "y(2) = 1 = 8a + 4b + 2c + 1, 8a + 4b + 2c = 0<BR>\n" );
document.write( "y(3) = 10 = 27a + 9b + 3c + 1, 27a + 9b + 3c = 9 --> 9a + 3b + c = 3\r<BR>\n" );
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document.write( " a + b  + c = -1<BR>\n" );
document.write( "8a + 4b + 2c = 0<BR>\n" );
document.write( "9a + 3b + c = 3\r<BR>\n" );
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document.write( "eq. 1 - eq. 3: a - 9a + b - 3b + c - c = -1 - 3<BR>\n" );
document.write( "-8a - 2b = -4<BR>\n" );
document.write( "-4a - b = -2<BR>\n" );
document.write( "4a + b = 2<BR>\n" );
document.write( "eq. 2 - 2*eq.3: 8a - 18a + 4b - 6b + 2c - 2c = 0 - 6<BR>\n" );
document.write( "-10a - 2b = -6<BR>\n" );
document.write( "-5a - b = -3<BR>\n" );
document.write( "5a + b = 3<BR>\n" );
document.write( "subtract 5a + b = 3 from 4a + b = 2:<BR>\n" );
document.write( "4a - 5a + b - b = 2 - 3<BR>\n" );
document.write( "-a = -1<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "substitute a = 1 into 4a + b = 2:<BR>\n" );
document.write( "4 + b = 2 --> b = -2<BR>\n" );
document.write( "subsitute a = 1 and b = -2 into a + b  + c = -1:<BR>\n" );
document.write( "1 - 2 + c = -1<BR>\n" );
document.write( "-1 + c = -1<BR>\n" );
document.write( "c = 0\r<BR>\n" );
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document.write( "cubic polynomial equation is:<BR>\n" );
document.write( "y = x^3 - 2x^2 + 1\r<BR>\n" );
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document.write( "y(0) = 1, yes<BR>\n" );
document.write( "y(1) = 1 - 2 + 1 = 0, yes<BR>\n" );
document.write( "y(2) = 8 - 8 + 1 = 1, yes<BR>\n" );
document.write( "y(3) = 27 - 18 + 1 = 9 + 1 = 10, yes\r<BR>\n" );
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document.write( "y(0.5) = (0.5)^3 - 2(0.5)^2 + 1<BR>\n" );
document.write( "y(0.5) = (1/2)^3 - 2(1/2)^2 + 1<BR>\n" );
document.write( "y(0.5) = 1/8 - 2 * 1/4 + 1<BR>\n" );
document.write( "y(0.5) = 1/8 - 2/4 + 1<BR>\n" );
document.write( "y(0.5) = 1/8 - 4/8 + 8/8<BR>\n" );
document.write( "y(0.5) = -3/8 + 8/8<BR>\n" );
document.write( "y(0.5) = 5/8<BR>\n" );
document.write( "y(0.5) = 0.625<BR>\n" );
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