document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191568>Question 1191568</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve -x^3 + 5x^2 - 8x + 4 ≥ 0 algebraically and graphically. \r<BR>\n" );
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document.write( "Please provide full answer. Thanks so much 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 #823366 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=823366\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Graph Approach:\r<BR>\n" );
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document.write( "If you typed \"graph -x^3 + 5x^2 - 8x + 4\" into google without quotes, then it will do as asked.<BR>\n" );
document.write( "Though the usual tool I use is GeoGebra. Another handy grapher is Desmos.<BR>\n" );
document.write( "There are tons of free options online.<BR>\n" );
document.write( "If you prefer your handheld graphing calculator, then of course go for that option.\r<BR>\n" );
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document.write( "Whichever graphing tool you use, a cubic curve results which looks like a sort of \"S\" shape in a sense.\r<BR>\n" );
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document.write( "The curve crosses the x axis at x = 1 and x = 2<BR>\n" );
document.write( "As the graph shows, when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3C+x+%3C+2 ALIGN=MIDDLE  ALT=\"1+%3C+x+%3C+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the curve is below the x axis.<BR>\n" );
document.write( "Also, when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+2 ALIGN=MIDDLE  ALT=\"x+%3E+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the curve is below the x axis.<BR>\n" );
document.write( "Otherwise, the graph is either on the x axis or above it.\r<BR>\n" );
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document.write( "So it's whenever <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+1 ALIGN=MIDDLE  ALT=\"x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or when <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;\" > is when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E3+%2B+5x%5E2+-+8x+%2B+4+%3E=0 ALIGN=MIDDLE  ALT=\"-x%5E3+%2B+5x%5E2+-+8x+%2B+4+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the case.\r<BR>\n" );
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document.write( "Algebraic Approach:\r<BR>\n" );
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document.write( "We'll use the rational root theorem.<BR>\n" );
document.write( "Since the first term is either -1 or +1, this means we can look at the plus/minus factors of the last term (4) to generate the list of all possible rational roots. <BR>\n" );
document.write( "That list is:<BR>\n" );
document.write( "-1, +1, -2, +2, -4, +4<BR>\n" );
document.write( "Of course something like +1 is the same as simply 1. I put the plus there to have it pair with its minus counterpart.\r<BR>\n" );
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document.write( "From here, we try all of these possible roots one at a time into the expression given.\r<BR>\n" );
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document.write( "Let's say we tried x = -1<BR>\n" );
document.write( "f(x) = -x^3 + 5x^2 - 8x + 4<BR>\n" );
document.write( "f(-1) = -(-1)^3 + 5(-1)^2 - 8(-1) + 4<BR>\n" );
document.write( "f(-1) = 18<BR>\n" );
document.write( "The result is not zero, so x = -1 is not a root of f(x)\r<BR>\n" );
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document.write( "On the other hand, x = 1 is a root because it does lead to f(x) = 0<BR>\n" );
document.write( "f(x) = -x^3 + 5x^2 - 8x + 4<BR>\n" );
document.write( "f(1) = -(1)^3 + 5(1)^2 - 8(1) + 4<BR>\n" );
document.write( "f(1) = 0\r<BR>\n" );
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document.write( "If you checked the others, you'd find that only x = 2 is the other root. In fact, it's a double root as the graph previously showed.<BR>\n" );
document.write( "This means the cubic factors to -(x-1)(x-2)^2\r<BR>\n" );
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document.write( "Once you know the roots are x = 1 and x = 2, you'll set up a number line with those values on it.<BR>\n" );
document.write( "Then pick something to the right of x = 1. Let's say we picked x = -1 since we already checked it and found that f(-1) = 18.<BR>\n" );
document.write( "This shows that if x < 1, then f(x) > 0\r<BR>\n" );
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document.write( "Now pick something between x = 1 and x = 2. Why not x = 1.5<BR>\n" );
document.write( "f(x) = -x^3 + 5x^2 - 8x + 4<BR>\n" );
document.write( "f(1.5) = -(1.5)^3 + 5(1.5)^2 - 8(1.5) + 4<BR>\n" );
document.write( "f(1.5) = -0.125<BR>\n" );
document.write( "The result is negative to indicate f(x) is negative on the interval 1 < x < 2, and this matches with what the graph shows.\r<BR>\n" );
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document.write( "Lastly, plug in something to the right of x = 2<BR>\n" );
document.write( "I'll pick x = 3<BR>\n" );
document.write( "f(x) = -x^3 + 5x^2 - 8x + 4<BR>\n" );
document.write( "f(3) = -(3)^3 + 5(3)^2 - 8(3) + 4<BR>\n" );
document.write( "f(3) = -2<BR>\n" );
document.write( "Therefore, f(x) < 0 when x > 2\r<BR>\n" );
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document.write( "Putting everything together, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E3+%2B+5x%5E2+-+8x+%2B+4+%3E=0 ALIGN=MIDDLE  ALT=\"-x%5E3+%2B+5x%5E2+-+8x+%2B+4+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > only true when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+1 ALIGN=MIDDLE  ALT=\"x+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or when <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;\" ><BR>\n" );
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