document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209690>Question 1209690</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all roots of the polynomial f(x) = 3x^3 - 7x^2 - 10x + 4 + 4x^3 - 5x^2 + 18x + 28.<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 #849819 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849819\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. **Combine like terms:**<BR>\n" );
document.write( "   f(x) = (3x^3 + 4x^3) + (-7x^2 - 5x^2) + (-10x + 18x) + (4 + 28)<BR>\n" );
document.write( "   f(x) = 7x^3 - 12x^2 + 8x + 32\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Rational Root Theorem:** This theorem helps us find possible rational roots.  Any rational root will be of the form p/q, where p is a factor of the constant term (32) and q is a factor of the leading coefficient (7).\r<BR>\n" );
document.write( "\n" );
document.write( "   Possible values for p: ±1, ±2, ±4, ±8, ±16, ±32<BR>\n" );
document.write( "   Possible values for q: ±1, ±7\r<BR>\n" );
document.write( "\n" );
document.write( "   Possible rational roots: ±1, ±2, ±4, ±8, ±16, ±32, ±1/7, ±2/7, ±4/7, ±8/7, ±16/7, ±32/7\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Test potential roots:** We can use synthetic division or direct substitution.  Let's try x = -2:\r<BR>\n" );
document.write( "\n" );
document.write( "   ```<BR>\n" );
document.write( "   -2 | 7  -12   8   32<BR>\n" );
document.write( "      | -14   52 -120<BR>\n" );
document.write( "      ------------------<BR>\n" );
document.write( "        7  -26  60 -88<BR>\n" );
document.write( "   ```\r<BR>\n" );
document.write( "\n" );
document.write( "   Since the remainder is not zero, -2 is not a root.\r<BR>\n" );
document.write( "\n" );
document.write( "   Let's try x = 4:\r<BR>\n" );
document.write( "\n" );
document.write( "   ```<BR>\n" );
document.write( "   4 | 7  -12   8   32<BR>\n" );
document.write( "     |  28   64  288<BR>\n" );
document.write( "     ------------------<BR>\n" );
document.write( "       7   16  72  320<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   Since the remainder is not zero, 4 is not a root.\r<BR>\n" );
document.write( "\n" );
document.write( "4. **Continue testing or use a calculator:**  Testing all the possible rational roots is tedious.  It's often more efficient to use a graphing calculator or online polynomial solver to approximate the roots.\r<BR>\n" );
document.write( "\n" );
document.write( "Using a calculator or solver, we find the roots are approximately:\r<BR>\n" );
document.write( "\n" );
document.write( "* x ≈ -1.089<BR>\n" );
document.write( "* x ≈ 1.402 + 1.494i<BR>\n" );
document.write( "* x ≈ 1.402 - 1.494i\r<BR>\n" );
document.write( "\n" );
document.write( "Notice that there is one real root and two complex roots. It's difficult to find these roots by hand. Numerical methods are the most practical approach.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );