document.write( "Question 940564: Write a polynomial function with rational coefficients in standard form with the given zeros. 2-i, square root of 5 \n" ); document.write( "

Algebra.Com's Answer #573391 by KMST(5328) $\"\"$ $\"About$

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If the polynomial must have real coefficients ,
\n" ); document.write( "having an imaginary root requires
\n" ); document.write( "also having the conjugate imaginary number as a root.
\n" ); document.write( "If the polynomial has an irrational root,
\n" ); document.write( "to have rational coefficients,
\n" ); document.write( "it must also have the conjugate irrational number as a root.
\n" ); document.write( "So your polynomial must have as roots
\n" ); document.write( " $\"2-i\"$ , $\"2%2Bi\"$ , $\"sqrt%285%29\"$ , and $\"-sqrt%285%29\"$ .
\n" ); document.write( "The simplest one would be
\n" ); document.write( "

\n" ); document.write( " $\"f%28x%29=%28x-2%2Bi%29%2A%28x-2-i%29%2A%28x%5E2-5%29\"$
\n" ); document.write( " $\"f%28x%29=%28%28x-2%29%5E2-i%5E2%29%2A%28x%5E2-5%29\"$
\n" ); document.write( " $\"f%28x%29=%28x%5E2-4x%2B4%2B1%29%2A%28x%5E2-5%29\"$
\n" ); document.write( " $\"f%28x%29=%28x%5E2-4x%2B5%29%2A%28x%5E2-5%29\"$
\n" ); document.write( " $\"f%28x%29=x%5E4-4x%5E3%2B20x-25\"$ \n" ); document.write( "

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