document.write( "Question 868265: Write. Polynomial function with rational coefficient in standard form with the given zeros x=3+i,2-i \n" ); document.write( "

Algebra.Com's Answer #523525 by KMST(5328) $\"\"$ $\"About$

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If a polynomial with rational coefficients has a non-real zero, it must also have the conjugate non-real zero.
\n" ); document.write( "That is to say that if $\"3%2Bi\"$ is a zero, $\"3-i\"$ must be a zero too.
\n" ); document.write( "That means that the factorization of the polynomial must contain the factors
\n" ); document.write( " $\"x-%283%2Bi%29=x-3-i=%28x-3%29-i\"$ and $\"x-%283-i%29=%28x-3%29%2Bi\"$ .
\n" ); document.write( "So the polynomial must have as a factor
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.
\n" ); document.write( "The same applied to the zero $\"2-i\"$ means that
\n" ); document.write( " $\"x-%282-i%29=x-2%2Bi=%28x-2%29%2Bi\"$ and $\"x-%282%2Bi%29=x-2-i=%28x-2%29-i\"$ area factors,
\n" ); document.write( "and the polynomial must have as a factor
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.
\n" ); document.write( "In sum, the polynomial must have for a factor
\n" ); document.write( " $\"%28x%5E2-6x%2B10%29%28x%5E2-4x%2B5%29=x%5E4-10x%5E3%2B39x%5E2-70x%2B50\"$ .
\n" ); document.write( "The simplest answer would be
\n" ); document.write( " $\"highlight%28f%28x%29=x%5E4-10x%5E3%2B39x%5E2-70x%2B50%29\"$ .
\n" ); document.write( "Of course, multiplying that polynomial by any rational number we can get another answer,
\n" ); document.write( "like $\"g%28x%29=3x%5E4-30x%5E3%2B117x%5E2-210x%2B150\"$ ,
\n" ); document.write( "or $\"g%28x%29=%282%2F3%29x%5E4-%2820%2F3%29x%5E3%2B26x%5E2-%28140%2F3%29x%2B100%2F3\"$ . \n" ); document.write( "

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