document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209684>Question 1209684</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all integers n for which \frac{n^2 + n + 1}{n - 2 + n^3} is an integer.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849816 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=849816\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Replace each instance of n with x.\r<BR>\n" );
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document.write( "The problem is<BR>\n" );
document.write( "<font color=blue>Find all integers x for which (x^2+x+1)/(x^3+x-2) is an integer</font>\r<BR>\n" );
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document.write( "Consider these two functions<BR>\n" );
document.write( "f(x) = abs(x^2+x+1)<BR>\n" );
document.write( "g(x) = abs(x^3+x-2)<BR>\n" );
document.write( "where \"abs\" is shorthand for \"absolute value\". <BR>\n" );
document.write( "Vertical bars can be used in place of abs notation.\r<BR>\n" );
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document.write( "The goal is to see when f/g is an integer for integer values of x.\r<BR>\n" );
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document.write( "Use a graphing tool of your choice, such as <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a>, to get this diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-2%2C20%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-2%2C20%2C-100%2Cabs%28x%5E2%2Bx%2B1%29%2Cabs%28x%5E3%2Bx-2%29%29%2C%0D%0Alocate%28-3.5%2C6%2Cf%28x%29%29%2C%0D%0Ablue%28locate%281.5%2C2%2Cg%28x%29%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "f(x) in green<BR>\n" );
document.write( "g(x) in blue<BR>\n" );
document.write( "Note that all of y = x^2+x+1 is entirely above the x axis, so abs(x^2+x+1) is identical to x^2+x+1. The absolute value bars have no effect on f(x).<BR>\n" );
document.write( "The abs is much more useful with g(x) since some parts of x^3+x-2 go below the x axis.\r<BR>\n" );
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document.write( "The two functions intersect at these approximate locations<BR>\n" );
document.write( "(0.3926, 1.5468)<BR>\n" );
document.write( "(1.8637, 6.3371)<BR>\n" );
document.write( "Which shows that the solutions to f(x) = g(x) are roughly x = 0.3926 and x = 1.8637\r<BR>\n" );
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document.write( "According to the graph, if 0.3926 < x < 1.8637 then the green curve is above the blue curve.<BR>\n" );
document.write( "This is when f(x) > g(x).<BR>\n" );
document.write( "Otherwise, f(x) < g(x)\r<BR>\n" );
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document.write( "So we only need to search for integers in the interval 0.3926 < x < 1.8637<BR>\n" );
document.write( "This narrow interval contains the integer x = 1 and no other integers.\r<BR>\n" );
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document.write( "f(x) = x^2+x+1<BR>\n" );
document.write( "f(1) = 1^2+1+1<BR>\n" );
document.write( "f(1) = 3<BR>\n" );
document.write( "g(x) = x^3+x-2<BR>\n" );
document.write( "g(1) = 1^3+1-2<BR>\n" );
document.write( "g(1) = 0<BR>\n" );
document.write( "h(x) = ( f(x) )/( g(x) )<BR>\n" );
document.write( "h(1) = ( f(1) )/( g(1) )<BR>\n" );
document.write( "h(1) = 3/0<BR>\n" );
document.write( "h(1) = undefined<BR>\n" );
document.write( "Recall we cannot divide by zero.\r<BR>\n" );
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document.write( "So f/g is not an integer, and not even defined, when x = 1.\r<BR>\n" );
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document.write( "If x is an integer other than 1, then f/g will not be an integer.<BR>\n" );
document.write( "Why not? Because the denominator g(x) is larger in absolute value compared to f(x).<BR>\n" );
document.write( "As an example, f/g = 7/8 when x = 2<BR>\n" );
document.write( "Another example is f/g = 3/(-12) = -1/4 when x = -2\r<BR>\n" );
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document.write( "So f/g is never an integer when x is an integer.<BR>\n" );
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