document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=149802>Question 149802</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.	Use synthetic division to divide the polynomial 2x^3 + 7x^2 – 5 by 3+x and write the quotient and the remainder\r<BR>\n" );
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document.write( "2.	Consider the polynomial f(x)= 4x^4+5x^3 + 7x^2 – 34x+8\r<BR>\n" );
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document.write( "By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.\r<BR>\n" );
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document.write( " 		Find all of the zeros of the given polynomial. Show procedure <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 #109921 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=109921\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1\r<BR>\n" );
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document.write( "Let's simplify this expression using synthetic division\r<BR>\n" );
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document.write( "Start with the given expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29 ALIGN=MIDDLE  ALT=\"%282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First lets find our test zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Bx=0 ALIGN=MIDDLE  ALT=\"3%2Bx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Bx ALIGN=MIDDLE  ALT=\"3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x.\r<BR>\n" );
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document.write( "so our test zero is -3\r<BR>\n" );
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document.write( "Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.(note: remember if a polynomial goes from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%5E2 ALIGN=MIDDLE  ALT=\"7x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5x%5E0 ALIGN=MIDDLE  ALT=\"-5x%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there is a zero coefficient for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E1 ALIGN=MIDDLE  ALT=\"x%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is simply because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+%2B+7x%5E2+-+5 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+7x%5E2+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > really looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3%2B7x%5E2%2B0x%5E1%2B-5x%5E0 ALIGN=MIDDLE  ALT=\"2x%5E3%2B7x%5E2%2B0x%5E1%2B-5x%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >7</TD><TD >0</TD><TD >-5</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></TR>\n" );
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document.write( "Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >7</TD><TD >0</TD><TD >-5</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -3 by 2 and place the product (which is -6)  right underneath the second  coefficient (which is 7)\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >7</TD><TD >0</TD><TD >-5</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-6</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -6 and 7 to get 1. Place the sum right underneath -6.\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >7</TD><TD >0</TD><TD >-5</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-6</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -3 by 1 and place the product (which is -3)  right underneath the third  coefficient (which is 0)\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >7</TD><TD >0</TD><TD >-5</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-6</TD><TD >-3</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -3 and 0 to get -3. Place the sum right underneath -3.\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >7</TD><TD >0</TD><TD >-5</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-6</TD><TD >-3</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD >-3</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -3 by -3 and place the product (which is 9)  right underneath the fourth  coefficient (which is -5)\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >7</TD><TD >0</TD><TD >-5</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-6</TD><TD >-3</TD><TD >9</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD >-3</TD><TD ></TD></TR>\n" );
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document.write( "    Add 9 and -5 to get 4. Place the sum right underneath 9.\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >7</TD><TD >0</TD><TD >-5</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-6</TD><TD >-3</TD><TD >9</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD >-3</TD><TD >4</TD></TR>\n" );
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document.write( "Since the last column adds to 4, we have a remainder of 4. This means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Bx ALIGN=MIDDLE  ALT=\"3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <b>not</b> a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+%2B+7x%5E2+-+5 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+7x%5E2+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets look at the bottom row of coefficients:\r<BR>\n" );
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document.write( "The first 3 coefficients (2,1,-3) form the quotient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+%2B+x+-+3 ALIGN=MIDDLE  ALT=\"2x%5E2+%2B+x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and the last coefficient 4, is the remainder, which is placed over <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Bx ALIGN=MIDDLE  ALT=\"3%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > like this\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%283%2Bx%29 ALIGN=MIDDLE  ALT=\"4%2F%283%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Putting this altogether, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+%2B+x+-+3%2B4%2F%283%2Bx%29 ALIGN=MIDDLE  ALT=\"2x%5E2+%2B+x+-+3%2B4%2F%283%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29=2x%5E2+%2B+x+-+3%2B4%2F%283%2Bx%29 ALIGN=MIDDLE  ALT=\"%282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29=2x%5E2+%2B+x+-+3%2B4%2F%283%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which looks like this in remainder form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29=2x%5E2+%2B+x+-+3 ALIGN=MIDDLE  ALT=\"%282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29=2x%5E2+%2B+x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > remainder 4\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the quotient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+%2B+x+-+3 ALIGN=MIDDLE  ALT=\"2x%5E2+%2B+x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the remainder is 4\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "Any rational zero can be found through this equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Roots=\frac{p}{q}\"> where p and q are the factors of the last and first coefficients\r<BR>\n" );
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document.write( "So let's list the factors of 8 (the last coefficient):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p=\pm1, \pm2, \pm4, \pm8\">\r<BR>\n" );
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document.write( "Now let's list the factors of 4 (the first coefficient):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q=\pm1, \pm2, \pm4\">\r<BR>\n" );
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document.write( "Now let's divide each factor of the last coefficient by each factor of the first coefficient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{1}, \frac{1}{2}, \frac{1}{4}, \frac{2}{1}, \frac{2}{2}, \frac{2}{4}, \frac{4}{1}, \frac{4}{2}, \frac{4}{4}, \frac{8}{1}, \frac{8}{2}, \frac{8}{4}, -\frac{1}{1}, -\frac{1}{2}, -\frac{1}{4}, -\frac{2}{1}, -\frac{2}{2}, -\frac{2}{4}, -\frac{4}{1}, -\frac{4}{2}, -\frac{4}{4}, -\frac{8}{1}, -\frac{8}{2}, -\frac{8}{4}\">\r<BR>\n" );
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document.write( "Now simplify\r<BR>\n" );
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document.write( "These are all the distinct rational zeros of the function that could occur\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  1, \frac{1}{2}, \frac{1}{4}, 2, 4, 8, -1, -\frac{1}{2}, -\frac{1}{4}, -2, -4, -8\">\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Now let's use synthetic division to test each possible zero:\r<BR>\n" );
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document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<tr><TD >1/2</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >2</TD><TD >7/2</TD><TD >21/4</TD><TD >-115/8</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >7</TD><TD >21/2</TD><TD >-115/4</TD><TD >-51/8</TD></tr></tr>\n" );
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document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-51%2F8 ALIGN=MIDDLE  ALT=\"-51%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<tr><TD >1/4</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >1</TD><TD >3/2</TD><TD >17/8</TD><TD >-255/32</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >6</TD><TD >17/2</TD><TD >-255/8</TD><TD >1/32</TD></tr></tr>\n" );
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document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F32 ALIGN=MIDDLE  ALT=\"1%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >2</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >8</TD><TD >26</TD><TD >66</TD><TD >64</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >13</TD><TD >33</TD><TD >32</TD><TD >72</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72 ALIGN=MIDDLE  ALT=\"72\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >4</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >16</TD><TD >84</TD><TD >364</TD><TD >1320</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >21</TD><TD >91</TD><TD >330</TD><TD >1328</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1328 ALIGN=MIDDLE  ALT=\"1328\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >8</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >32</TD><TD >296</TD><TD >2424</TD><TD >19120</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >37</TD><TD >303</TD><TD >2390</TD><TD >19128</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19128 ALIGN=MIDDLE  ALT=\"19128\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >-1</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-4</TD><TD >-1</TD><TD >-6</TD><TD >40</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >1</TD><TD >6</TD><TD >-40</TD><TD >48</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48 ALIGN=MIDDLE  ALT=\"48\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >-1/2</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-2</TD><TD >-3/2</TD><TD >-11/4</TD><TD >147/8</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >3</TD><TD >11/2</TD><TD >-147/4</TD><TD >211/8</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=211%2F8 ALIGN=MIDDLE  ALT=\"211%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F4 ALIGN=MIDDLE  ALT=\"-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >-1/4</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-1</TD><TD >-1</TD><TD >-3/2</TD><TD >71/8</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >4</TD><TD >6</TD><TD >-71/2</TD><TD >135/8</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=135%2F8 ALIGN=MIDDLE  ALT=\"135%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F4 ALIGN=MIDDLE  ALT=\"-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >-2</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-8</TD><TD >6</TD><TD >-26</TD><TD >120</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-3</TD><TD >13</TD><TD >-60</TD><TD >128</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=128 ALIGN=MIDDLE  ALT=\"128\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >-4</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-16</TD><TD >44</TD><TD >-204</TD><TD >952</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-11</TD><TD >51</TD><TD >-238</TD><TD >960</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=960 ALIGN=MIDDLE  ALT=\"960\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >-8</TD><TD >|</TD><TD >4</TD><TD >5</TD><TD >7</TD><TD >-34</TD><TD >8</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-32</TD><TD >216</TD><TD >-1784</TD><TD >14544</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-27</TD><TD >223</TD><TD >-1818</TD><TD >14552</TD></tr></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14552 ALIGN=MIDDLE  ALT=\"14552\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 ALIGN=MIDDLE  ALT=\"4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "====================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since none of the possible rational roots are actual roots, this means that the polynomial either has irrational roots or complex roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );