document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=725393>Question 725393</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Give (a) the domain and (b) the zeros of f.\r<BR>\n" );
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document.write( "f(x) = 3x^3 - x^2 - 2x / x(x+1)^2\r<BR>\n" );
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document.write( "Thank you for your help! :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #444532 by </B><A HREF=/tutors/aboutme.mpl?userid=General_Lee87><B>General_Lee87(58)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=General_Lee87><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=General_Lee87><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=444532\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The domain appears to be all values of x except x = -1 and x = 0<BR>\n" );
document.write( " The range appears to be all values of y.<BR>\n" );
document.write( " The zeroes (roots) appear to be x = -2/3 and x = 1. \r<BR>\n" );
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document.write( "you find the domain by looking for all values of x that will result in a value of y.<BR>\n" );
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document.write( "all values of x will result in a value of y except when x = -1 and when x = 0. \r<BR>\n" );
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document.write( "when x = -1 or when x = 0, the denominator of the original equation will be equal to 0 which makes the value of y undefined.<BR>\n" );
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document.write( "therefore x = -1 and x = 0 can't be in the domain. \r<BR>\n" );
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document.write( "since there are no further restrictions as to what the value of x can be, then the domain is all values of x except at x = -1 and x = 0.<BR>\n" );
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document.write( "the zeroes of the equation are found by setting the equation equal to 0 and solving for x.<BR>\n" );
document.write( " it's best to simplify the equation first. \r<BR>\n" );
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document.write( "start with (3x^3 - x^2 - 2x) / (x * (x+1)^2 \r<BR>\n" );
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document.write( "since (3x^3 - x^2 - 2x) / x is equal to (3x^2 - x - 2), the equation becomes: \r<BR>\n" );
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document.write( "(3x^2 - x - 2) / (x+1)^2 \r<BR>\n" );
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document.write( "set this equal to 0 to get: \r<BR>\n" );
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document.write( "(3x^2 - x - 2) / (x + 1)^2 = 0 \r<BR>\n" );
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document.write( "multiply both sides of the equation by (x + 1)^2 to get: \r<BR>\n" );
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document.write( "(3x^2 - x - 2) = 0 \r<BR>\n" );
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document.write( "factor to get: \r<BR>\n" );
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document.write( "(3x + 2) (x - 1) = 0 \r<BR>\n" );
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document.write( "solve for x to get: \r<BR>\n" );
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document.write( "x = -2/3 or x = 1 \r<BR>\n" );
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document.write( "these are the roots of the quadratic equation which is the point at which the equation crosses the x-axis which is the zeroes of the equation.<BR>\n" );
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document.write( "you can confirm that the solutions are correct by graphing the equation. \r<BR>\n" );
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document.write( "from the graph, you can see that the zeroes of the graph are around x = -2/3 and x = 1.<BR>\n" );
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document.write( "from the graph, you can see that there appears to be an asymptote at around x = -1 although it's not real easy to see that.<BR>\n" );
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document.write( "from the graph, you cannot see that there is a hole at x = 0 \r<BR>\n" );
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document.write( "the value of y is undefined at x = -1 and also undefined at x = 0. \r<BR>\n" );
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document.write( "this is because the denominator of the equation is 0 at those values of x. \r<BR>\n" );
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document.write( "at x = -1, this results in an asymptote. \r<BR>\n" );
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document.write( "at x = 0, this results in a hole. \r<BR>\n" );
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document.write( "if you try to find the value of y when x = 0 and when x = -1, you will not be able to.<BR>\n" );
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document.write( "you can find a value of y for any other value of x other than 0 or -1. </SPAN>\n" );
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