document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=152515>Question 152515</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many real zeroes are there in f(x) = (x + 3)(x – 4)(x + 6)(x^2 + 7)? <BR>\n" );
document.write( "@ This is fairly straightforward. Only the last factor results in imaginary solutions  <BR>\n" );
document.write( "a. 3 <BR>\n" );
document.write( "b. 5 <BR>\n" );
document.write( "c. 4 <BR>\n" );
document.write( "d. None of the above <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #112185 by </B><A HREF=/tutors/aboutme.mpl?userid=mducky2><B>mducky2(62)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mducky2><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=112185\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>Let's find the zeroes of the function. Don't be fooled by the terminology. A zero of the function merely means any value of x that makes f(x) = 0. If any of the factors are zero, f(x) would equal zero. Therefore, we should individually solve for x with each factor equated to zero.<BR>\n" );
document.write( "<br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+3+=+0 ALIGN=MIDDLE  ALT=\"x+%2B+3+=+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+=+-3 ALIGN=MIDDLE  ALT=\"x+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since x is a real number, this looks like a real zero.<BR>\n" );
document.write( "<br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+4+=+0 ALIGN=MIDDLE  ALT=\"x+-+4+=+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+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since x is a real number, this also looks like a real zero.<BR>\n" );
document.write( "<br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+6+=+0 ALIGN=MIDDLE  ALT=\"x+%2B+6+=+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+=+-6 ALIGN=MIDDLE  ALT=\"x+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since x is a real number, this still looks like a real zero.<BR>\n" );
document.write( "<br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+7+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+7+=+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+=+-7 ALIGN=MIDDLE  ALT=\"x%5E2+=+-7\"   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%28-7%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%28-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Any square root of a negative number will not be a real number, so this is not a real zero.<BR>\n" );
document.write( "<br>It looks like there are 3 total zeroes. We know there can't be any more because we have equated every possible factor of the function with zero. Therefore, the answer is a. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );