document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=558544>Question 558544</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f(x)=5x^4 -7x^2 +4.\r<BR>\n" );
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document.write( "a. How many zeros does f(x) have?\r<BR>\n" );
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document.write( "b. Based on the graph, how many real number zeros does f(x) have?\r<BR>\n" );
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document.write( "I know the answers are a. 4 and b.2 but I don't understand how you get that. Can you explain please? These are from my exam review packet. Help !! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #363094 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=363094\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. I agree that it should have 4 complex zeros, because it is a polynomial of degree 4.<BR>\n" );
document.write( "b. Why the reference to a graph? Are you supposed to use a graphing calculator?<BR>\n" );
document.write( "Are you supposed to calculate minima and maxima based on derivatives?<BR>\n" );
document.write( "A graphing calculator would show you somthing like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1.5%2C1.5%2C-2%2C14%2C5x%5E4-7x%5E2%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1.5%2C1.5%2C-2%2C14%2C5x%5E4-7x%5E2%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > showing that there are no real zeros.<BR>\n" );
document.write( "You could graph with just pencil and paper, based on the derivative, if you knew a little calculus. You could analyze the function easily enough.<BR>\n" );
document.write( "The function is an even function, symmetrical with respect to the y-axis, meaning that f(x)=f(-x). It is obvious that for x with large absolute value, the function is positive, and it grows without bounds towards both ends (towards -infinity and +infinity).<BR>\n" );
document.write( "A little algebra can transform the function into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=5%28x%5E2-7%2F10%29%5E2%2B31%2F20 ALIGN=MIDDLE  ALT=\"f%28x%29=5%28x%5E2-7%2F10%29%5E2%2B31%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which tells you that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29%3E=31%2F20 ALIGN=MIDDLE  ALT=\"f%28x%29%3E=31%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The minima will occur when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=7%2F10 ALIGN=MIDDLE  ALT=\"x%5E2=7%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you will have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=31%2F20 ALIGN=MIDDLE  ALT=\"f%28x%29=31%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" > then.<BR>\n" );
document.write( "You could also try to solve<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E4-7x%5E2%2B4=0 ALIGN=MIDDLE  ALT=\"5x%5E4-7x%5E2%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > by changing variables with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find that there are no real solutions for y, so there are no rela solutions for x either. </SPAN>\n" );
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