document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=96495>Question 96495</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a problem that asks me to find the one real zero and two complex zeros. I just do not know where to even begin and our book has no examples of it. Here is the problem: 2x^3+17x^2+120x+225  Thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #70264 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=70264\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm not sure how your teacher wants you to find the first root, but we'll use a graphing calculator to find the first root.\r<BR>\n" );
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document.write( "Using the root find feature, we find one root at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5%2F2 ALIGN=MIDDLE  ALT=\"x=-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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+17x%5E2+%2B+120x+%2B+225%29%2F%282x%2B5%29 ALIGN=MIDDLE  ALT=\"%282x%5E3+%2B+17x%5E2+%2B+120x+%2B+225%29%2F%282x%2B5%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=2x%2B5=0 ALIGN=MIDDLE  ALT=\"2x%2B5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B5 ALIGN=MIDDLE  ALT=\"2x%2B5\"   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=-5%2F2 ALIGN=MIDDLE  ALT=\"x=-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x.\r<BR>\n" );
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document.write( "so our test zero is -5/2\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.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-5/2</TD><TD >|</TD><TD >2</TD><TD >17</TD><TD >120</TD><TD >225</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 >-5/2</TD><TD >|</TD><TD >2</TD><TD >17</TD><TD >120</TD><TD >225</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 -5/2 by 2 and place the product (which is -5)  right underneath the second  coefficient (which is 17)\r<BR>\n" );
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document.write( "<TR><TD >-5/2</TD><TD >|</TD><TD >2</TD><TD >17</TD><TD >120</TD><TD >225</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</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 -5 and 17 to get 12. Place the sum right underneath -5.\r<BR>\n" );
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document.write( "<TR><TD >-5/2</TD><TD >|</TD><TD >2</TD><TD >17</TD><TD >120</TD><TD >225</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >12</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -5/2 by 12 and place the product (which is -30)  right underneath the third  coefficient (which is 120)\r<BR>\n" );
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document.write( "<TR><TD >-5/2</TD><TD >|</TD><TD >2</TD><TD >17</TD><TD >120</TD><TD >225</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-30</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >12</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -30 and 120 to get 90. Place the sum right underneath -30.\r<BR>\n" );
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document.write( "<TR><TD >-5/2</TD><TD >|</TD><TD >2</TD><TD >17</TD><TD >120</TD><TD >225</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-30</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >12</TD><TD >90</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -5/2 by 90 and place the product (which is -225)  right underneath the fourth  coefficient (which is 225)\r<BR>\n" );
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document.write( "<TR><TD >-5/2</TD><TD >|</TD><TD >2</TD><TD >17</TD><TD >120</TD><TD >225</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-30</TD><TD >-225</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >12</TD><TD >90</TD><TD ></TD></TR>\n" );
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document.write( "    Add -225 and 225 to get 0. Place the sum right underneath -225.\r<BR>\n" );
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document.write( "<TR><TD >-5/2</TD><TD >|</TD><TD >2</TD><TD >17</TD><TD >120</TD><TD >225</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-30</TD><TD >-225</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >12</TD><TD >90</TD><TD >0</TD></TR>\n" );
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document.write( "Since the last column adds to zero, we have a remainder of zero. This means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B5 ALIGN=MIDDLE  ALT=\"2x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+%2B+17x%5E2+%2B+120x+%2B+225 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+17x%5E2+%2B+120x+%2B+225\"   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 (2,12,90) form the quotient\r<BR>\n" );
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document.write( "Notice in the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B5 ALIGN=MIDDLE  ALT=\"2x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x term has a coefficient of 2, so we need to divide the quotient by 2 like this:\r<BR>\n" );
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document.write( "  \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+17x%5E2+%2B+120x+%2B+225%29%2F%282x%2B5%29=x%5E2+%2B+6x+%2B+45 ALIGN=MIDDLE  ALT=\"%282x%5E3+%2B+17x%5E2+%2B+120x+%2B+225%29%2F%282x%2B5%29=x%5E2+%2B+6x+%2B+45\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can use this <a  rel=nofollow HREF=http://calc101.com/webMathematica/long-divide.jsp>online polynomial division calculator</a> to check your work\r<BR>\n" );
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document.write( "Basically  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+%2B+17x%5E2+%2B+120x+%2B+225 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+17x%5E2+%2B+120x+%2B+225\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B5%29%28x%5E2+%2B+6x+%2B+45%29 ALIGN=MIDDLE  ALT=\"%282x%2B5%29%28x%5E2+%2B+6x+%2B+45%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets break  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+45 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+45\"   DISABLED_style= \"max-width:90%; height:auto;\" > down further \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( "Starting with the general quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\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%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6%2Ax%2B45=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B6%2Ax%2B45=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <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=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=45 ALIGN=MIDDLE  ALT=\"c=45\"   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+=+%28-6+%2B-+sqrt%28+%286%29%5E2-4%2A1%2A45+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+%286%29%5E2-4%2A1%2A45+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=6, and c=45\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+sqrt%28+36-4%2A1%2A45+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+36-4%2A1%2A45+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 6 to get 36  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+sqrt%28+36%2B-180+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+36%2B-180+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A45%2A1 ALIGN=MIDDLE  ALT=\"-4%2A45%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-180 ALIGN=MIDDLE  ALT=\"-180\"   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+=+%28-6+%2B-+sqrt%28+-144+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+-144+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+12%2Ai%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+12%2Ai%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, 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+=+%28-6+%2B-+12%2Ai%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+12%2Ai%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 1 to get 2\r<BR>\n" );
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document.write( "After simplifying, the quadratic has roots of\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3+%2B+6%2Ai ALIGN=MIDDLE  ALT=\"x=-3+%2B+6%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3+-+6%2Ai ALIGN=MIDDLE  ALT=\"x=-3+-+6%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So that means the polynomial has roots of \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5%2F2 ALIGN=MIDDLE  ALT=\"x=-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is real), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3+%2B+6%2Ai ALIGN=MIDDLE  ALT=\"x=-3+%2B+6%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3+-+6%2Ai ALIGN=MIDDLE  ALT=\"x=-3+-+6%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > (and the last two are complex) </SPAN>\n" );
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