document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=709135>Question 709135</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do you find all of the real zeroes of 8x^3+125 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #436462 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=436462\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3%2B125 ALIGN=MIDDLE  ALT=\"8x%5E3%2B125\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To find the zeroes of an expression of degree 3 or more (like this one) we need to factor it.<br><BR>\n" );
document.write( "When factoring, check the greatest common factor (GCF) first. The GCF here is 1 (which we rarely factor out).<br><BR>\n" );
document.write( "After the GCF, we look to use any and all of the other factoring techniques. One of these techniques is factoring by the use of factoring patterns. And one of these patterns (the only one in fact) that has two terms with a \"+\" between them is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3%2Bb%5E3+=+%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"a%5E3%2Bb%5E3+=+%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As the pattern shows, we need a sum of two cubes to use this pattern. Is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3%2B125 ALIGN=MIDDLE  ALT=\"8x%5E3%2B125\"   DISABLED_style= \"max-width:90%; height:auto;\" > a sum of cubes? It's definitely a sum. Are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3 ALIGN=MIDDLE  ALT=\"8x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125 ALIGN=MIDDLE  ALT=\"125\"   DISABLED_style= \"max-width:90%; height:auto;\" > perfect cubes? With a little investigation we should find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3+=+%282x%29%5E3 ALIGN=MIDDLE  ALT=\"8x%5E3+=+%282x%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125+=+5%5E3 ALIGN=MIDDLE  ALT=\"125+=+5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we can use the pattern with the \"a\" being \"2x\" and the \"b\" being \"5\". So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3%2B125 ALIGN=MIDDLE  ALT=\"8x%5E3%2B125\"   DISABLED_style= \"max-width:90%; height:auto;\" >, according to the patterns factors into:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%282x%29%2B%285%29%29%28%282x%29%5E2-%282x%29%285%29%2B%285%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28%282x%29%2B%285%29%29%28%282x%29%5E2-%282x%29%285%29%2B%285%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Note: My use of parentheses may seem excessive. But it is actually a good idea to use parentheses like this when making substitutions like we have done here. It helps us avoid easily-made mistakes.)<BR>\n" );
document.write( "This simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B5%29%284x%5E2-10x%2B25%29 ALIGN=MIDDLE  ALT=\"%282x%2B5%29%284x%5E2-10x%2B25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Neither factor will factor further no matter which techniques we try.<br><BR>\n" );
document.write( "From the Zero Product Property we know that this product will be zero <i>only</i> be zero if one of the factors is zero. And if this product is zero then the expression from which we got these factors, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3%2B125 ALIGN=MIDDLE  ALT=\"8x%5E3%2B125\"   DISABLED_style= \"max-width:90%; height:auto;\" > will also be zero. So the x values that make these factors zero will be the zeroes of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3%2B125 ALIGN=MIDDLE  ALT=\"8x%5E3%2B125\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Setting each factor to zero and then solving will lead to our solution. For the first factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B5+=+0 ALIGN=MIDDLE  ALT=\"2x%2B5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving this we should get x = -5/2.<br><BR>\n" );
document.write( "For the second factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-10x%2B25+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2-10x%2B25+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We must use the Quadratic Formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-10%29+%2B-+sqrt%28%28-10%29%5E2-4%284%29%2825%29%29%29%2F2%284%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-10%29+%2B-+sqrt%28%28-10%29%5E2-4%284%29%2825%29%29%29%2F2%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-10%29+%2B-+sqrt%28100-4%284%29%2825%29%29%29%2F2%284%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-10%29+%2B-+sqrt%28100-4%284%29%2825%29%29%29%2F2%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-10%29+%2B-+sqrt%28100-400%29%29%2F2%284%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-10%29+%2B-+sqrt%28100-400%29%29%2F2%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-10%29+%2B-+sqrt%28-300%29%29%2F2%284%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-10%29+%2B-+sqrt%28-300%29%29%2F2%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point we can stop. The negative in the square root means that the solutions will be complex numbers. But the problem specifically asks for real solutions. So the only real zero to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3%2B125 ALIGN=MIDDLE  ALT=\"8x%5E3%2B125\"   DISABLED_style= \"max-width:90%; height:auto;\" > is -5/2.<br><BR>\n" );
document.write( "P.S. In case you are curious about the complex zeroes I will go ahead and finish with the Quadratic Formula. Continuing the simplifying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+sqrt%28-300%29%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+sqrt%28-300%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+sqrt%28-1%2A100%2A3%29%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+sqrt%28-1%2A100%2A3%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+sqrt%28-1%29%2Asqrt%28100%29%2Asqrt%283%29%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+sqrt%28-1%29%2Asqrt%28100%29%2Asqrt%283%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+i%2A10%2Asqrt%283%29%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+i%2A10%2Asqrt%283%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%285+%2B-+5i%2Asqrt%283%29%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%282%285+%2B-+5i%2Asqrt%283%29%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285+%2B-+5i%2Asqrt%283%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%285+%2B-+5i%2Asqrt%283%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is short for:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285+%2B+5i%2Asqrt%283%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%285+%2B+5i%2Asqrt%283%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285+-+5i%2Asqrt%283%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%285+-+5i%2Asqrt%283%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In standard \"a + bi\" form these would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5%2F4+%2B+%285sqrt%283%29%2F4%29i ALIGN=MIDDLE  ALT=\"x+=+5%2F4+%2B+%285sqrt%283%29%2F4%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5%2F4+%2B+%28-5sqrt%283%29%2F4%29i ALIGN=MIDDLE  ALT=\"x+=+5%2F4+%2B+%28-5sqrt%283%29%2F4%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So there are three zeroes: two complex (above) and the one real we found earlier (-5/2). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );