document.write( "Question 1134148: Indentify the number of positive, negative, and imaginary roots, all rational roots , and the total amount of roots.
\n" ); document.write( "x^3+x-6=0 \n" ); document.write( "

Algebra.Com's Answer #751613 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "The polynomial on the left side is of degree 3 - so the total amount of the roots is 3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It includes real and complex number roots.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The derivative of the left side polynomial is  3x^2 + 1; from the first glance,\r\n" );
document.write( "\r\n" );
document.write( "it is clear that this derivative function is always positive - so the original \r\n" );
document.write( "\r\n" );
document.write( "polynomial function on the left side is monotonically increasing.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It means, that there is ONLY ONE real root.  Hence, other two roots are complex numbers.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At x= 0, the left side function has the value of -6, and it rises monotonically to +infinity as  x --> oo   - \r\n" );
document.write( "\r\n" );
document.write( "    hence, the real root is a positive real number. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then, according to the Rational Root theorem, rational roots can be only among these numbers (divisors of 6): 1, 2, 3 or 6.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "But easy mental check shows that the number 2 is not the root, and, moreover, at x= 2 the left side function is just positive, \r\n" );
document.write( "so the only real root is IRRATIONAL number between 1 and 2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the required analysis is COMPLETED, and its results are as follows:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    - total number of roots is 3;\r\n" );
document.write( "\r\n" );
document.write( "    - there is only one real root;\r\n" );
document.write( "\r\n" );
document.write( "    - there are two complex roots;\r\n" );
document.write( "\r\n" );
document.write( "    - the real root is positive;\r\n" );
document.write( "\r\n" );
document.write( "    - the number of positive real roots is 1;\r\n" );
document.write( "\r\n" );
document.write( "    - the number of negative real roots is zero;\r\n" );
document.write( "\r\n" );
document.write( "    - the number of rational real roots is zero.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "All the questions are answered; so the solution is completed.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );