document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=185101>Question 185101</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/061849281X/algebrahomeworkh>college algebra concepts and models</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> find all real zeros of the function algebraically ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E5%2Bx%5E3-6x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E5%2Bx%5E3-6x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "analyze the graph of the function algebraically ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=-x%5E2%2B10x-16 ALIGN=MIDDLE  ALT=\"g%28x%29=-x%5E2%2B10x-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #138853 by </B><A HREF=/tutors/aboutme.mpl?userid=ZeratuelX><B>ZeratuelX(5)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ZeratuelX><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=138853\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Finding all real zeros means that we are finding all possible values for x that will satisfy the equation and still are in the \"real number system\".\r<BR>\n" );
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document.write( "If we have the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E5%2Bx%5E3-6x ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E5%2Bx%5E3-6x\"   DISABLED_style= \"max-width:90%; height:auto;\" > we are looking for all of the x-intercepts, which are the same as roots/zeros. The best way to find these algebraically is to factor it out. In this function there is an x we can factor out from the beginning.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%2A%28x%5E4%2Bx%5E2-6%29 ALIGN=MIDDLE  ALT=\"y+=+x%2A%28x%5E4%2Bx%5E2-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we are looking for 2 numbers that add up to be 1, and multiply to be -6. The two numbers I came up with are -2 and 3. Let's factor this out using the reverse FOIL method and these numbers.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%2A%28x%5E2%2B3%29%2A%28x%5E2-2%29 ALIGN=MIDDLE  ALT=\"y+=+x%2A%28x%5E2%2B3%29%2A%28x%5E2-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since we are finding the zeros, which are essentially roots/x-intercepts, we can set y = 0 because the x-intercept will have a y coordinate of 0. We now have this equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+x%2A%28x%5E2%2B3%29%2A%28x%5E2-2%29 ALIGN=MIDDLE  ALT=\"0+=+x%2A%28x%5E2%2B3%29%2A%28x%5E2-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now by the zero factor theorem, if the product is 0, than we can set each factor to 0.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2+=+0 ALIGN=MIDDLE  ALT=\"x%5E2-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we solve for x in each of these.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solved.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+-3 ALIGN=MIDDLE  ALT=\"x%5E2+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%28-3%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 2 Answers.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%28-3%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solved.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2+=+0 ALIGN=MIDDLE  ALT=\"x%5E2-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+2 ALIGN=MIDDLE  ALT=\"x%5E2+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%282%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 2 Answers.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%282%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solved.\r<BR>\n" );
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document.write( "The first equation is already solved, we already have 1 real root. The second equation can't be done in the real number system since we cannot take the square root of a negative number without using the imaginary number system. The third equation has two possible values, both of which are real and satisfy the equation.<BR>\n" );
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document.write( "In the original equation we had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5 ALIGN=MIDDLE  ALT=\"x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since it is raised to the 5th power, there are 5 total solutions, and we have found 3 of them to be real zeroes. The other two are imaginary numbers (from case 2). So our answer would be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%282%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Note that these numbers are irrational, but still real.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%282%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > This is because they represent a decimal, that is between two numbers.<BR>\n" );
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document.write( "For your second question, we need to graph it and describe the graph. So the equation would look like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=-x%5E2%2B10x-16 ALIGN=MIDDLE  ALT=\"g%28x%29=-x%5E2%2B10x-16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+330%2C+300%2C+-1%2C+10%2C+-10%2C+10%2C+-x%5E2%2B10x-16%29+ ALIGN=MIDDLE  ALT=\"+graph%28+330%2C+300%2C+-1%2C+10%2C+-10%2C+10%2C+-x%5E2%2B10x-16%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can see that the graph opens down. Since it is a parabola, it starts going up, but it hits a vertex at (5,9) and then starts to go down. </SPAN>\n" );
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