document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199775>Question 199775</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi!\r<BR>\n" );
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document.write( "Find all real zeros of the polynomial. Use the quadratic formula if necessary.\r<BR>\n" );
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document.write( "P(x) = 4x^3 - 7x^2 - 10x - 2\r<BR>\n" );
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document.write( "thanks for the homework help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150137 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=150137\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, let's find the possible rational zeros of P(x):\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 -2 (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\">\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{-1}{1}, \frac{-1}{2}, \frac{-1}{4}, \frac{-2}{1}, \frac{-2}{2}, \frac{-2}{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, -1, \frac{-1}{2}, \frac{-1}{4}, -2\">\r<BR>\n" );
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document.write( "Now let's test the possible rational zeros to see which ones are actually roots.\r<BR>\n" );
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document.write( "Let's see if 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;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   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" );
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document.write( "<tr><TD >1</TD><TD >|</TD><TD >4</TD><TD >-7</TD><TD >-10</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >4</TD><TD >-3</TD><TD >-13</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-3</TD><TD >-13</TD><TD >-15</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=-15 ALIGN=MIDDLE  ALT=\"-15\"   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%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's see if 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;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   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 >-7</TD><TD >-10</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >2</TD><TD >-5/2</TD><TD >-25/4</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-5</TD><TD >-25/2</TD><TD >-33/4</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=-33%2F4 ALIGN=MIDDLE  ALT=\"-33%2F4\"   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%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's see if 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;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   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 >-7</TD><TD >-10</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >1</TD><TD >-3/2</TD><TD >-23/8</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-6</TD><TD >-23/2</TD><TD >-39/8</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=-39%2F8 ALIGN=MIDDLE  ALT=\"-39%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%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's see if 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;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   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" );
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document.write( "<tr><TD >2</TD><TD >|</TD><TD >4</TD><TD >-7</TD><TD >-10</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >8</TD><TD >2</TD><TD >-16</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >1</TD><TD >-8</TD><TD >-18</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=-18 ALIGN=MIDDLE  ALT=\"-18\"   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%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's see if 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;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   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" );
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document.write( "<tr><TD >-1</TD><TD >|</TD><TD >4</TD><TD >-7</TD><TD >-10</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-4</TD><TD >11</TD><TD >-1</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-11</TD><TD >1</TD><TD >-3</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=-3 ALIGN=MIDDLE  ALT=\"-3\"   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%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's see if 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;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   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" );
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document.write( "<tr><TD >-1</TD><TD >|</TD><TD >4</TD><TD >-7</TD><TD >-10</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-4</TD><TD >11</TD><TD >-1</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-11</TD><TD >1</TD><TD >-3</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=-3 ALIGN=MIDDLE  ALT=\"-3\"   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%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's see if 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;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   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 >-7</TD><TD >-10</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-2</TD><TD >9/2</TD><TD >11/4</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-9</TD><TD >-11/2</TD><TD >3/4</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=3%2F4 ALIGN=MIDDLE  ALT=\"3%2F4\"   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%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "------------------------------------------------------\r<BR>\n" );
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document.write( "Let's see if 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;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   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( "<TABLE  cellpadding=10>\n" );
document.write( "<tr><TD >-1/4</TD><TD >|</TD><TD >4</TD><TD >-7</TD><TD >-10</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-1</TD><TD >2</TD><TD >2</TD></tr><tr><TD ></TD><TD ></TD><TD >4</TD><TD >-8</TD><TD >-8</TD><TD >0</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=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is 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 a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3-7x%5E2-10x-2=%284x%2B1%29%28x%5E2-2x-2%29 ALIGN=MIDDLE  ALT=\"4x%5E3-7x%5E2-10x-2=%284x%2B1%29%28x%5E2-2x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: the quotient results from taking a quarter of the first three values in the bottom row.\r<BR>\n" );
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document.write( "Now let's solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x-2=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the next two zeros.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x-2=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "Notice we have a quadratic in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%5E2%2BBx%2BC ALIGN=MIDDLE  ALT=\"Ax%5E2%2BBx%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1 ALIGN=MIDDLE  ALT=\"A=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-2 ALIGN=MIDDLE  ALT=\"B=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=-2 ALIGN=MIDDLE  ALT=\"C=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for \"x\":\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 ALIGN=MIDDLE  ALT=\"x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-2%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-2%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1 ALIGN=MIDDLE  ALT=\"A=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-2 ALIGN=MIDDLE  ALT=\"B=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=-2 ALIGN=MIDDLE  ALT=\"C=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-2%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-2%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+4-4%281%29%28-2%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4-4%281%29%28-2%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+4--8+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4--8+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281%29%28-2%29 ALIGN=MIDDLE  ALT=\"4%281%29%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+4%2B8+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4%2B8+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284--8%29 ALIGN=MIDDLE  ALT=\"sqrt%284--8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%2B8%29 ALIGN=MIDDLE  ALT=\"sqrt%284%2B8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+12+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+12+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+12+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+12+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+2%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+2%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root  (note: If you need help with simplifying square roots, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%29%2F%282%29+%2B-+%282%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282%29%2F%282%29+%2B-+%282%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1+%2B-+sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+1+%2B-+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+1%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1-sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+1-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression.  \r<BR>\n" );
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document.write( "So the next two roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+1%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1-sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+1-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "========================================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the three real zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+4x%5E3+-+7x%5E2+-+10x+-+2 ALIGN=MIDDLE  ALT=\"P%28x%29+=+4x%5E3+-+7x%5E2+-+10x+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F4 ALIGN=MIDDLE  ALT=\"x=-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+1%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1-sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+1-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "If you have any questions, email me at jim_thompson5910@hotmail.com </SPAN>\n" );
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