document.write( "Question 1039745: Can I please have your assistance? What are the least, and most, amount of distinct zeroes of a 7th degree polynomial, given that at least one root is a complex number? \n" ); document.write( "

Algebra.Com's Answer #654481 by robertb(5830) $\"\"$ $\"About$

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If at least one of the roots is a complex number, then its complex conjugate is also a root of the polynomial. If the complex root has multiplicity 3, then the complex conjugate root would also have multiplicity 3. It would follow that exactly one root would be real, and the 7th degree polynomial would have the form $\"alpha%2A%28x%5E2%2Bax%2Bb%29%5E3%2A%28x-d%29\"$ ,
\n" ); document.write( "where d is the lone real root. In this case the least number of distinct roots is 3.\r
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\n" ); document.write( "\n" ); document.write( "It is also possible that the polynomial would have three distinct complex roots. If this is the case then the form of the polynomial would be
\n" ); document.write( " $\"alpha%2A%28x%5E2%2Bax%2Bb%29%28x%5E2%2Bcx%2Bd%29%28x%5E2%2Bex%2Bf%29%2A%28x-g%29\"$ ,
\n" ); document.write( "in which case there would be 7 distinct zeroes, the most number of distinct zeroes. (g is the lone real root.)\r
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