document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208426>Question 1208426</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use synthetic division to determine if <BR>\n" );
document.write( "x - c is a factor of the given polynomial. \r<BR>\n" );
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document.write( "3x^4 + x^3 - 3x + 1; [x + (1/3)]\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846838 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=846838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Think of 3x^4+x^3-3x+1 as 3x^4+1x^3+0x^2-3x+1<BR>\n" );
document.write( "The coefficients from left to right are 3,1,0,-3,1<BR>\n" );
document.write( "Write those coefficients along the top row of the synthetic division table. To the left of these coefficients is the test root -1/3. This is from solving x + 1/3 = 0 to get x = -1/3.\r<BR>\n" );
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document.write( "This is the set up.<BR>\n" );
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document.write( "<tr><TD >-1/3</TD><TD ></TD><TD >3</TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD >1</TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "Then we'll pull down the first coefficient (3)<BR>\n" );
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document.write( "<tr><TD >-1/3</TD><TD ></TD><TD >3</TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD >1</TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD ></TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "Multiply that with the test root (-1/3). <BR>\n" );
document.write( "(-1/3)*3 = -1<BR>\n" );
document.write( "Place the result under the next coefficient.\r<BR>\n" );
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document.write( "<tr><TD >-1/3</TD><TD ></TD><TD >3</TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD >1</TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD >-1</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD ></TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "Then add straight down: 1 + (-1) = 0<BR>\n" );
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document.write( "<tr><TD >-1/3</TD><TD ></TD><TD >3</TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD >1</TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD >-1</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "We'll repeat this process of multiplying with the test root and adding straight down to get this next column of values.<BR>\n" );
document.write( "(-1/3)*0 = 0<BR>\n" );
document.write( "0+0 = 0<BR>\n" );
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document.write( "<tr><TD >-1/3</TD><TD ></TD><TD >3</TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD >1</TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD >-1</TD><TD >0</TD><TD ></TD><TD ></TD></tr><tr><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "Repeat the process again.<BR>\n" );
document.write( "(-1/3)*0 = 0<BR>\n" );
document.write( "-3+0 = -3<BR>\n" );
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document.write( "<tr><TD >-1/3</TD><TD ></TD><TD >3</TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD >1</TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD ></TD></tr><tr><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD >0</TD><TD >-3</TD><TD ></TD></tr>\n" );
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document.write( "Then one last time.<BR>\n" );
document.write( "(-1/3)*(-3) = 1<BR>\n" );
document.write( "1+1 = <font color=red>2</font><BR>\n" );
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document.write( "<tr><TD >-1/3</TD><TD ></TD><TD >3</TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD >1</TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >1</TD></tr><tr><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD >0</TD><TD >-3</TD><TD ><font color=red size=4>2</font></TD></tr>\n" );
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document.write( "The value in the bottom right corner is the remainder. I have highlighted it in <font color=red>red</font>\r<BR>\n" );
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document.write( "The nonzero remainder means that x+(1/3) is not a factor of 3x^4 + x^3 - 3x + 1\r<BR>\n" );
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document.write( "Here's another method that doesn't involve synthetic division.\r<BR>\n" );
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document.write( "If x-c is a factor of f(x), then f(c) = 0. This is a special case of the remainder theorem.<BR>\n" );
document.write( "Rewrite x + (1/3) as x - (-1/3) to determine c = -1/3.\r<BR>\n" );
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document.write( "We'll plug this into the polynomial to see if we get zero or not.<BR>\n" );
document.write( "f(x) = 3x^4 + x^3 - 3x + 1<BR>\n" );
document.write( "f(-1/3) = 3(-1/3)^4 + (-1/3)^3 - 3(-1/3) + 1<BR>\n" );
document.write( "f(-1/3) = 3(1/81) - 1/27 + 1 + 1<BR>\n" );
document.write( "f(-1/3) = 1/27 - 1/27 + 1 + 1<BR>\n" );
document.write( "f(-1/3) = 2<BR>\n" );
document.write( "The nonzero result tells us that x+(1/3) is not a factor of 3x^4 + x^3 - 3x + 1.<BR>\n" );
document.write( "Note: The result 2 is the remainder we got in the previous section.\r<BR>\n" );
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document.write( "Yet another method involves graphing y = 3x^4 + x^3 - 3x + 1.<BR>\n" );
document.write( "Notice how the curve doesn't pass through the x axis when x = -1/3.<BR>\n" );
document.write( "This visually confirms our answer above.<BR>\n" );
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