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 #823374 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823374\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "To find where the value of the expression is positive (or negative), we want to find the values for which it is 0.<br><BR>\n" );
document.write( "For an algebraic approach, start with the rational root theorem which tells us the possible rational roots are 1, -1, 2, -2, 4, and -4.<br><BR>\n" );
document.write( "Checking to see if 1 is a root is easy simply by substituting x=1 in the expression.  In this example, we quickly see that 1 is a root: -1+5-8+4=0.<br><BR>\n" );
document.write( "x=1 is a root, so (x-1) is a factor of the expression.  Reduce the expression to a quadratic by removing the factor of (x-1) using long division or synthetic division.<br><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( "  1 | -1  5 -8  4\r\n" );
document.write( "    |    -1  4 -4\r\n" );
document.write( "    +-------------\r\n" );
document.write( "      -1  4 -4  0</PRE><BR>\n" );
document.write( "The reduced polynomial is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2%2B4x-4+=+-1%28x%5E2-4x%2B4%29+=+-1%28x-2%29%28x-2%29 ALIGN=MIDDLE  ALT=\"-x%5E2%2B4x-4+=+-1%28x%5E2-4x%2B4%29+=+-1%28x-2%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the completely factored polynomial is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E3%2B5x%5E2-8x%2B4=%28-1%29%28x-1%29%28x-2%29%28x-2%29 ALIGN=MIDDLE  ALT=\"-x%5E3%2B5x%5E2-8x%2B4=%28-1%29%28x-1%29%28x-2%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The zeros of the expression are 1 and 2; those are the only values of x where, as you \"walk\" along the x axis, the value of the function can change from positive to negative or vice versa.<br><BR>\n" );
document.write( "To answer the question, we need to find the values of x for which the value of the expression is 0 or positive.<br><BR>\n" );
document.write( "To do that, we can \"walk\" along the x axis.  The way I find easiest to do that is to start with a \"large\" positive value of x and walk along the x-axis to the left.<br><BR>\n" );
document.write( "The factored form of the expression is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%28x-1%29%28x-2%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28-1%29%28x-1%29%28x-2%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >}.<br><BR>\n" );
document.write( "For \"large\" values of x (anything greater than the largest root, 2), the signs of the factors are -+++, so the product is negative.  So the given inequality is NOT satisfied for large values of x.<br><BR>\n" );
document.write( "Walking left along the x-axis, nothing changes until we reach the largest root, 2.  When we pass x=2, the signs of TWO factors change at the same time, so the sign of the expression does not change.  The signs of the factors are -+--, so the product is still negative.  Since the given inequality is for greater than OR EQUAL TO 0, the inequality is satisfied at the single point x=2.<br><BR>\n" );
document.write( "Continuing our walk to the left, when we pass the next root at x=1, the sign of a single factor changes, so the signs of the factors are now ----, and the product is positive, and the given inequality is satisfied.  And x=1 is the smallest root, so the inequality is satisfied for x=1 and for all values of x less than 1.<br><BR>\n" );
document.write( "Our final answer is that the value of the given expression is greater than or equal to zero for <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;\" > and for the single value x=2.<br><BR>\n" );
document.write( "ANSWER (in interval notation): (-infinity,1] U [2,2]<br><BR>\n" );
document.write( "A graph showing that solution set:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-3%2C3%2C-2%2C2%2C-x%5E3+%2B+5x%5E2+-+8x+%2B+4%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-3%2C3%2C-2%2C2%2C-x%5E3+%2B+5x%5E2+-+8x+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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