document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92193>Question 92193</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0395977258/algebrahomeworkh>Algebra and Trigonometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> please help me understand and solve the equation x^3 + 2x^2 + x + 2 = 0, if -2 is a root. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #67030 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=67030\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If -2 is a root, then -2 is a test zero. So that means we can use synthetic division\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 >-2</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD >2</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 1)\r<BR>\n" );
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document.write( "<TR><TD >-2</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -2 by 1 and place the product (which is -2)  right underneath the second  coefficient (which is 2)\r<BR>\n" );
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document.write( "<TR><TD >-2</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD >2</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 >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -2 and 2 to get 0. Place the sum right underneath -2.\r<BR>\n" );
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document.write( "<TR><TD >-2</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD >2</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 >1</TD><TD >0</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -2 by 0 and place the product (which is 0)  right underneath the third  coefficient (which is 1)\r<BR>\n" );
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document.write( "<TR><TD >-2</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD >0</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 0 and 1 to get 1. Place the sum right underneath 0.\r<BR>\n" );
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document.write( "<TR><TD >-2</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD >0</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -2 by 1 and place the product (which is -2)  right underneath the fourth  coefficient (which is 2)\r<BR>\n" );
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document.write( "<TR><TD >-2</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD >0</TD><TD >-2</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD ></TD></TR>\n" );
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document.write( "    Add -2 and 2 to get 0. Place the sum right underneath -2.\r<BR>\n" );
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document.write( "<TR><TD >-2</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-2</TD><TD >0</TD><TD >-2</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</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=x%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B+2x%5E2+%2B+x+%2B+2 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+2x%5E2+%2B+x+%2B+2\"   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 (1,0,1) 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=%28x%5E3+%2B+2x%5E2+%2B+x+%2B+2%29%2F%28x%2B2%29=x%5E2+%2B+1 ALIGN=MIDDLE  ALT=\"%28x%5E3+%2B+2x%5E2+%2B+x+%2B+2%29%2F%28x%2B2%29=x%5E2+%2B+1\"   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=x%5E3+%2B+2x%5E2+%2B+x+%2B+2 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+2x%5E2+%2B+x+%2B+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%28x%5E2+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28x%5E2+%2B+1%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+1 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+1\"   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%2B1=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: since the polynomial does not have an \"x\" term, the 2nd coefficient is zero. In other words, b=0. So that means the polynomial really looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B0%2Ax%2B1=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B0%2Ax%2B1=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=0 ALIGN=MIDDLE  ALT=\"b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=1 ALIGN=MIDDLE  ALT=\"c=1\"   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+=+%280+%2B-+sqrt%28+%280%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+sqrt%28+%280%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=0, and c=1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%280+%2B-+sqrt%28+0-4%2A1%2A1+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+sqrt%28+0-4%2A1%2A1+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 0 to get 0  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%280+%2B-+sqrt%28+0%2B-4+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+sqrt%28+0%2B-4+%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%2A1%2A1 ALIGN=MIDDLE  ALT=\"-4%2A1%2A1\"   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+=+%280+%2B-+sqrt%28+-4+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+sqrt%28+-4+%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+=+%280+%2B-+2%2Ai%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+2%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+=+%280+%2B-+2%2Ai%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+2%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( "Answer:\r<BR>\n" );
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document.write( "So the polynomial has roots \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=i ALIGN=MIDDLE  ALT=\"x=i\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-i ALIGN=MIDDLE  ALT=\"x=-i\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the last two are the imaginary roots) </SPAN>\n" );
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