document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=148299>Question 148299</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (22 pts) Consider the polynomial f(x) = 2x^3 – 3x^2 – 8x – 3.\r<BR>\n" );
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document.write( "(i) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.\r<BR>\n" );
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document.write( "(ii) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.\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 #108699 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=108699\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i)\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 -3 (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, \pm3\">\r<BR>\n" );
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document.write( "Now let's list the factors of 2 (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\">\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{3}{1}, \frac{3}{2}, \frac{-1}{1}, \frac{-1}{2}, \frac{-3}{1}, \frac{-3}{2}\">\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}, 3, \frac{3}{2}, -1, \frac{-1}{2}, -3, \frac{-3}{2}\">\r<BR>\n" );
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document.write( "--------------------------------------------------------\r<BR>\n" );
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document.write( "ii)\r<BR>\n" );
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document.write( "With the help of a graphing calculator, we see that -1 is a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-3x%5E2-8x-3 ALIGN=MIDDLE  ALT=\"2x%5E3-3x%5E2-8x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "note: let me know if you need to find the zeros a different way.\r<BR>\n" );
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document.write( "So let's set up a synthetic division table by placing the value -1 in the upper left corner and placing the coefficients of the polynomial to the right of -1.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-1</TD><TD >|</TD><TD >2</TD><TD >-3</TD><TD >-8</TD><TD >-3</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 >-1</TD><TD >|</TD><TD >2</TD><TD >-3</TD><TD >-8</TD><TD >-3</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 -1 by 2 and place the product (which is -2)  right underneath the second  coefficient (which is -3)\r<BR>\n" );
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document.write( "<TR><TD >-1</TD><TD >|</TD><TD >2</TD><TD >-3</TD><TD >-8</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</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 -2 and -3 to get -5. Place the sum right underneath -2.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-1</TD><TD >|</TD><TD >2</TD><TD >-3</TD><TD >-8</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-5</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -1 by -5 and place the product (which is 5)  right underneath the third  coefficient (which is -8)\r<BR>\n" );
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document.write( "<TR><TD >-1</TD><TD >|</TD><TD >2</TD><TD >-3</TD><TD >-8</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD >5</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-5</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 5 and -8 to get -3. Place the sum right underneath 5.\r<BR>\n" );
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document.write( "<TR><TD >-1</TD><TD >|</TD><TD >2</TD><TD >-3</TD><TD >-8</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD >5</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-5</TD><TD >-3</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -1 by -3 and place the product (which is 3)  right underneath the fourth  coefficient (which is -3)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-1</TD><TD >|</TD><TD >2</TD><TD >-3</TD><TD >-8</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD >5</TD><TD >3</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-5</TD><TD >-3</TD><TD ></TD></TR>\n" );
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document.write( "    Add 3 and -3 to get 0. Place the sum right underneath 3.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-1</TD><TD >|</TD><TD >2</TD><TD >-3</TD><TD >-8</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD >5</TD><TD >3</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-5</TD><TD >-3</TD><TD >0</TD></TR>\n" );
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document.write( "Since the last column adds to zero, this means that -1 is a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-3x%5E2-8x-3 ALIGN=MIDDLE  ALT=\"2x%5E3-3x%5E2-8x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this confirms our original claim). \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,-5,-3) form the quotient\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3+-+3x%5E2+-+8x+-+3%29%2F%28x%2B1%29=2x%5E2+-+5x+-+3 ALIGN=MIDDLE  ALT=\"%282x%5E3+-+3x%5E2+-+8x+-+3%29%2F%28x%2B1%29=2x%5E2+-+5x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Basically  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+-+3x%5E2+-+8x+-+3 ALIGN=MIDDLE  ALT=\"2x%5E3+-+3x%5E2+-+8x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%282x%5E2+-+5x+-+3%29 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%282x%5E2+-+5x+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets find the zeros for  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+5x+-+3 ALIGN=MIDDLE  ALT=\"2x%5E2+-+5x+-+3\"   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-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%282%29%28-3%29+%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%282%29%28-3%29+%29%29%2F%282%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-5 ALIGN=MIDDLE  ALT=\"b=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-3 ALIGN=MIDDLE  ALT=\"c=-3\"   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+=+%285+%2B-+sqrt%28+49+%29%29%2F%284%29 ALIGN=MIDDLE  ALT=\"x+=+%285+%2B-+sqrt%28+49+%29%29%2F%284%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=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+=+%285+%2B-+7%29%2F%284%29 ALIGN=MIDDLE  ALT=\"x+=+%285+%2B-+7%29%2F%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49 ALIGN=MIDDLE  ALT=\"49\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   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+=+%2812%29%2F%284%29 ALIGN=MIDDLE  ALT=\"x+=+%2812%29%2F%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=++%28-2%29%2F%284%29 ALIGN=MIDDLE  ALT=\"x+=++%28-2%29%2F%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \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;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1%2F2 ALIGN=MIDDLE  ALT=\"x+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. \r<BR>\n" );
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document.write( "So the zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-3x%5E2-8x-3 ALIGN=MIDDLE  ALT=\"2x%5E3-3x%5E2-8x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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