document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209717>Question 1209717</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f(x) be a polynomial.  Find the remainder when f(x) is divided by x(x - 1)(x - 2), if f(0) = 0, f(1) = 1, and f(2) = 2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849935 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849935\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the remainder:\r<BR>\n" );
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document.write( "1. **The Remainder Theorem:**  When a polynomial f(x) is divided by (x - c), the remainder is f(c).  We're given f(0) = 0, f(1) = 1, and f(2) = 2.\r<BR>\n" );
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document.write( "2. **Form of the Remainder:** Since we're dividing by a cubic (x(x - 1)(x - 2)), the remainder will be a polynomial of degree at most 2. Let the remainder be R(x) = ax² + bx + c.\r<BR>\n" );
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document.write( "3. **Set up equations:** We know the following:\r<BR>\n" );
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document.write( "*   R(0) = f(0) = 0<BR>\n" );
document.write( "*   R(1) = f(1) = 1<BR>\n" );
document.write( "*   R(2) = f(2) = 2\r<BR>\n" );
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document.write( "Substitute these values into R(x):\r<BR>\n" );
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document.write( "*   R(0) = a(0)² + b(0) + c = 0  => c = 0<BR>\n" );
document.write( "*   R(1) = a(1)² + b(1) + c = 1  => a + b + c = 1<BR>\n" );
document.write( "*   R(2) = a(2)² + b(2) + c = 2  => 4a + 2b + c = 2\r<BR>\n" );
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document.write( "4. **Solve for a and b:**  We already know c = 0.  Substitute this into the other two equations:\r<BR>\n" );
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document.write( "*   a + b = 1<BR>\n" );
document.write( "*   4a + 2b = 2\r<BR>\n" );
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document.write( "Notice that the second equation is just 2 times the first equation. This means the two equations are dependent and there are infinitely many solutions for a and b.\r<BR>\n" );
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document.write( "Let's use the first equation to express b in terms of a:<BR>\n" );
document.write( "b = 1 - a\r<BR>\n" );
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document.write( "Substitute this into the second equation:<BR>\n" );
document.write( "4a + 2(1-a) = 2<BR>\n" );
document.write( "4a + 2 - 2a = 2<BR>\n" );
document.write( "2a = 0<BR>\n" );
document.write( "a = 0\r<BR>\n" );
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document.write( "Then, b = 1-a = 1-0 = 1.\r<BR>\n" );
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document.write( "Therefore, a = 0 and b = 1.\r<BR>\n" );
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document.write( "5. **Write the remainder:**  Since a = 0, b = 1, and c = 0, the remainder is:\r<BR>\n" );
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document.write( "R(x) = 0x² + 1x + 0 = x\r<BR>\n" );
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document.write( "Therefore, the remainder when f(x) is divided by x(x - 1)(x - 2) is simply *x*.<BR>\n" );
document.write( " </SPAN>\n" );
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