document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=191251>Question 191251</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780395977255/algebrahomeworkh>algebra and trigonometry structure and method book 2</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of times r is a root of P(x)=0.<BR>\n" );
document.write( "P(x)= x^4+4x^3-16x-16; r=-2\r<BR>\n" );
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document.write( "i found the depressed eruation twice but the numbers were all messed up. please help and explain. </SPAN>\n" );
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document.write( "If you found the depressed equation twice, you should be at an easy to solve quadratic.\r<BR>\n" );
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document.write( "Check your work based on the following results when I performed the polynomial long division:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{P(x)}{x - (-2)} = \frac{x^4+4x^3-16x-16}{x + 2} = x^2 +2x^2 -4x -8\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{x^2 +2x^2 -4x -8}{x + 2} = x^2 - 4\">\r<BR>\n" );
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document.write( "And\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 4 = (x + 2)(x - 2)\">, hence the roots are \r<BR>\n" );
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document.write( "-2 (with a multiplicity of 3) and 2.\r<BR>\n" );
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document.write( "I used polynomial long division, but synthetic division would work just as well.  A common error when performing these division operations is failing to explictly specify all orders of the variable in the dividend.  In other words, you have to divide into:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^4 + 4x^3 + 0x^2 - 16x - 16\">\r<BR>\n" );
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document.write( "or if you are using synthetic division, your first line of coefficients must look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\  4\  0\  -16\  -16\">\r<BR>\n" );
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document.write( "Another possible source of error would be if you were dividing by x - 2.  Remember, if the root is -2, then the factor is x + 2, and that should have been your divisor.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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