document.write( "Question 564982: I am trying to help a student at a residential treatment center. We are looking for the number and type of complex solutions and possible real solutions for the following:\r
\n" ); document.write( "\n" ); document.write( "2x^2+5x+3=0\r
\n" ); document.write( "\n" ); document.write( "4x^3-12x+9=0\r
\n" ); document.write( "\n" ); document.write( "2x^4+x^2-x+6=0 \n" ); document.write( "

Algebra.Com's Answer #365742 by richard1234(7193) $\"\"$ $\"About$

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The number of complex solutions of a polynomial is always equal to the degree of the polynomial (this is called the fundamental theorem of algebra).\r
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\n" ); document.write( "\n" ); document.write( "You can find the possible rational roots quite easily using the rational root theorem. The possible rational roots of a polynomial are in the form

where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem can easily be proven using modular arithmetic.\r
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\n" ); document.write( "\n" ); document.write( "There isn't much of a way to find possible \"real\" roots. However, you do know that if P(a) is negative and P(b) is positive, there exists at least one real zero between a and b. This is due to the intermediate value theorem, which states that for a continuous function f(x) between a and b, every number between f(a) and f(b) has at least one x-value in the domain. \n" ); document.write( "

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