document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171009>Question 1171009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I don't need the solution to the following question. I just need someone to explain to me what is the question!...more of a language problem. If after understanding the question and if I am unable to solve the problem I will repost the question.\r<BR>\n" );
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document.write( "Q:  Write the polynomial equation of lowest degree with 'constant (real or complex) coefficients' having the roots 4 and 2-3i.\r<BR>\n" );
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document.write( "What is a constant coefficient? Constant in the polynomial is the coefficient of x^0. With the given, the factors are (x-4),(x-2+3i),(x-2-3i)<BR>\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #795921 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52879)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=795921\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
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document.write( "x^2 + 3x - 4  is a polynomial of the degree 2 with constant coefficients  \r\n" );
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document.write( "      1 (at x^2),  3 (at x)  and  -4 (the constant term).\r\n" );
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document.write( "x^3 - 6x^2 + 7x - 1  is a polynomial of the degree 3 with constant coefficients  \r\n" );
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document.write( "      1 (at x^3),  -6 (at x^2),  7 (at x)  and -1 (the constant term).\r\n" );
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document.write( "-2x4 - 14x^3 + 5x^2 + 19x + 3  is a polynomial of the degree 4 with constant coefficients  \r\n" );
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document.write( "      -2 (at x^4),  -14 (at x^3),  5 (at x^2),  19 (at x)  and  3 (the constant term).\r\n" );
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document.write( "After looking into these examples, it should be clear to you what the term \"coefficients of a polynomial\"\r\n" );
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document.write( "or the \"constant coefficients of a polynomial\" means.\r\n" );
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document.write( "The coefficients in the examples above are integer numbers.\r\n" );
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document.write( "But they can be any real numbers or even complex numbers.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "I hope that after my explanations the subject is a bit more clear to you.\r<BR>\n" );
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document.write( "If your interests on the subject go/stretch farther, you may read this Wikipedia article \r<BR>\n" );
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document.write( "https://en.wikipedia.org/wiki/Polynomial\r<BR>\n" );
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document.write( "Comment from student: Ref: Question 1171009 Thank you for your help. I did look up the reference you provided as well. <BR>\n" );
document.write( "In the context of polynomials, I am familiar with what the coefficient means and what Integral, Real, Rational, <BR>\n" );
document.write( "and complex coefficients mean. I am still unclear about the \"Constant coefficient\". Is it just another <BR>\n" );
document.write( "or complete name for the Coefficients of the variables that can be Integral, Real, Rational or complex? <BR>\n" );
document.write( "If that is so, then the answer (please ref the question that I posed) in the book goes as follows: <BR>\n" );
document.write( "(x-2)[x-(1-3i)]=0 So what happened to [x-(1+3i)] ? I suspect it has to do with the word 'Constant'? Please help.\r<BR>\n" );
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document.write( "<U>My response</U> : Thank you for asking.\r<BR>\n" );
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document.write( "First, I am very satisfied that you do refer to the Problem's ID number.\r<BR>\n" );
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document.write( "It tells me that you are an responsible person.\r<BR>\n" );
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document.write( "Next, couple of words about the terminology.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "    When we consider polynomials, there are two major entities: there is variable \" x \" with its degrees; it is considered\r\n" );
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document.write( "    (in accordance with its name) as variable value, which may have different values;\r\n" );
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document.write( "    and there are \"coefficients\" that are considered as constant values for each individual polynomial.\r\n" );
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document.write( "Next issue is about the polynomial from the Problem's question and about its factors.\r<BR>\n" );
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document.write( "    The question is :  \"Q: Write the polynomial equation of lowest degree with 'constant (real or complex) coefficients' \r\n" );
document.write( "    having the roots 4 and 2-3i.\"\r\n" );
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document.write( "    There are two VERY DIFFERENT, I'd say, \"opposite\" cases here:  \r\n" );
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document.write( "        a) when the requested polynomial is considered over the set of complex numbers (\"with complex coefficients\"), and \r\n" );
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document.write( "        b) when the polynomial is considered over the set of real numbers (\"with real coefficients\").\r\n" );
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document.write( "    The polynomial of the LOWEST degree with COMPLEX coefficients (case a) having the given roots 4 and 2-3i\r\n" );
document.write( "    is the polynomial  (x-4)*(x-(2-3i)).  It is the product of two binomials associated with the roots. \r\n" );
document.write( "    It is the quadratic polynomial, and if you want to write it in the standard form, you should multiply \r\n" );
document.write( "    these two linear binomial with complex coefficients (making FOIL).\r\n" );
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document.write( "    The polynomial of the LOWEST degree with REAL coefficients (case b) having the roots  4 and 2-3i, NECESSARY\r\n" );
document.write( "    has the complex conjugate root to the number 2-3i, which is 2+3i.\r\n" );
document.write( "    Therefore, in this case, such a polynomial has, actually, three roots 4, 2-3i and 2+3i.\r\n" );
document.write( "    And as such, it is the polynomial of the 3rd degree (x-4)*(x-(2-3i))*(x-(2+3i)).\r\n" );
document.write( "    It is the product of THREE binomials associated with the THREE roots.\r\n" );
document.write( "    Again, to write it in the standard form, you should multiply (FOIL) these 3 binomials.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "These are the facts from Algebra.\r<BR>\n" );
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document.write( "These facts are proved in courses of Abstract Algebra of the University level.\r<BR>\n" );
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document.write( "For high school students these facts only announced (without formal proofs).\r<BR>\n" );
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document.write( "So, it seems to me that I answered all your questions.\r<BR>\n" );
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document.write( "As you understand, it is not my goal to bring/(to provide) a full course of Abstract Algebra here,<BR>\n" );
document.write( "so I should to limit myself by these short explanations.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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