document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=73678>Question 73678</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am trying to find all real and imaginary zeros for the equations:  (9x^3+9x^2-19x-7) &  (7x^3+5x^2+26x-8)  I am more interested in finding how to solve then the solutions themselves.  I can find no factors to divide into the equations!  I am usually very good at these but was given these in class and can't find out how to solve them?!  VERY Frustrating!  Thanx </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #53195 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=53195\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Well you sure did pick a hard problem to tackle, I'll see what I can do to help. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E3%2B9x%5E2-19x-7=0 ALIGN=MIDDLE  ALT=\"9x%5E3%2B9x%5E2-19x-7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >Start with the given polynomial<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E3%2B9x%5E2-18x-x-6-1=0 ALIGN=MIDDLE  ALT=\"9x%5E3%2B9x%5E2-18x-x-6-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >Rewrite the polynomial in terms that are in common with the rest of the polynomial. For instance, 18 is divisible by 6. These tricks are somewhat difficult since there are no set rules for when or how you use them.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289x%5E3%2B9x%5E2%29%2B%28-18x-6%29%2B%28-x-1%29=0 ALIGN=MIDDLE  ALT=\"%289x%5E3%2B9x%5E2%29%2B%28-18x-6%29%2B%28-x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >Start grouping common terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2%28x%2B1%29-6%283x%2B1%29-%28x%2B1%29=0 ALIGN=MIDDLE  ALT=\"9x%5E2%28x%2B1%29-6%283x%2B1%29-%28x%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >Factor out GCF of each grouped term. Notice how we have a common factor of x+1. We can subtract like terms (it's like saying y=x+1 and 9x^2y-y=(9x^2-1)y)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2%28x%2B1%29-%28x%2B1%29-6%283x%2B1%29=0 ALIGN=MIDDLE  ALT=\"9x%5E2%28x%2B1%29-%28x%2B1%29-6%283x%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289x%5E2-1%29%28x%2B1%29-6%283x%2B1%29=0 ALIGN=MIDDLE  ALT=\"%289x%5E2-1%29%28x%2B1%29-6%283x%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now factor the (9x^2-1) term to get (3x+1)(3x-1) notice the common term of 3x+1. Add these like terms.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B1%29%283x-1%29%28x%2B1%29-6%283x%2B1%29=0 ALIGN=MIDDLE  ALT=\"%283x%2B1%29%283x-1%29%28x%2B1%29-6%283x%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283x-1%29%28x%2B1%29-6%29%283x%2B1%29=0 ALIGN=MIDDLE  ALT=\"%28%283x-1%29%28x%2B1%29-6%29%283x%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >So if you use synthetic division,  you will find that 3x+1 is a factor and x=-1/3 is a zero. You can use this information to find the other zeros. When you use synthetic division, you will get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B2x-7 ALIGN=MIDDLE  ALT=\"3x%5E2%2B2x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is a factor (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E2%2B2x-7%29%283x%2B1%29=9x%5E3%2B9x%5E2-19x-7 ALIGN=MIDDLE  ALT=\"%283x%5E2%2B2x-7%29%283x%2B1%29=9x%5E3%2B9x%5E2-19x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now use the quadratic formula on 3x^2+2x-7 to find the other zeros<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B2x%2B-7+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B2x%2B-7+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%282%29%5E2-4%2A3%2A-7=88 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%282%29%5E2-4%2A3%2A-7=88\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=88 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-2%2B-sqrt%28+88+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-2%2B-sqrt%28+88+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%282%29%2Bsqrt%28+88+%29%29%2F2%5C3+=+1.23013858660781 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%282%29%2Bsqrt%28+88+%29%29%2F2%5C3+=+1.23013858660781\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%282%29-sqrt%28+88+%29%29%2F2%5C3+=+-1.89680525327448 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%282%29-sqrt%28+88+%29%29%2F2%5C3+=+-1.89680525327448\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B2x%2B-7 ALIGN=MIDDLE  ALT=\"3x%5E2%2B2x%2B-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B2x%2B-7+=+3%28x-1.23013858660781%29%2A%28x--1.89680525327448%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B2x%2B-7+=+3%28x-1.23013858660781%29%2A%28x--1.89680525327448%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 1.23013858660781, -1.89680525327448.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B2%2Ax%2B-7+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B2%2Ax%2B-7+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "So your zeros are x=-1/3, x=1.23014, and x=-1.89681 <BR>\n" );
document.write( "Feel free to ask about any step. Hope this helps. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );