document.write( "Question 1085271: Please write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 2,2i, 4-[6] as zeros. The brackets [] are for square roots.
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Algebra.Com's Answer #699348 by KMST(5328)\"\" \"About 
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TIP: The conventional way to write \"sqrt%286%29\" is sqrt(6).
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\n" ); document.write( "1) For each real or complex zero \"p\" of a polynomial \"f%28x%29\" ,
\n" ); document.write( "there is a factor \"%28x-p%29\" in the factorization of the polynomial.
\n" ); document.write( "So, if \"f%28x%29\" has \"2\" , \"2i\", and \"4-sqrt%286%29\" as zeros,
\n" ); document.write( "the factors \"%28x-2%29\" , \"%28x-2i%29\" , and \"%28x-%284-sqrt%286%29%29%29=%28x-4%2Bsqrt%286%29%29\" are in the factorization of \"f%28x%29\" .
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\n" ); document.write( "2) If a polynomial \"f%28x%29\" has real coefficients and has a non-real complex number \"b%2Bi%2Ac\" for example) as a zero,
\n" ); document.write( "the conjugate complex number, \"b-i%2Ac\" is also a zero of the polynomial,
\n" ); document.write( "and
\n" ); document.write( "if \"f%28x%29\" has rational coefficients and has an irrational number such as \"d-e%2Asqrt%28f%29\" as a zero,
\n" ); document.write( "the conjugate irrational number, \"d%2Be%2Asqrt%28f%29\" , is also a zero of the polynomial.
\n" ); document.write( "In other words, irrational zeros and non-real complex zeros come as conjugate pairs.
\n" ); document.write( "So, along with \"2i\" , \"f%28x%29\" also has \"-2i\" as a zero,
\n" ); document.write( "and along with \"4-sqrt%286%29\" , \"f%28x%29\" also has \"4%2Bsqrt%286%29\" as a zero.
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\n" ); document.write( "Then, the factors \"%28x-%28-2i%29%29=%28x%2B2i%29\" and
\n" ); document.write( "\"%28x-%284%2Bsqrt%286%29%29%29=%28x-4-sqrt%286%29%29\" are also in the factorization of \"f%28x%29\" .
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\n" ); document.write( "3) Besides at least one factor of the form \"%28x-p%29\" for every zero,
\n" ); document.write( "the factorization of a polynomial with rational coefficients,
\n" ); document.write( "also has a constant factor \"a\" , which will be the leading coefficient.
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\n" ); document.write( "So,
\n" ); document.write( "\"f%28x%29=1%28x-2%29%28x-2i%29%28x%2B2i%29%28x-4%2Bsqrt%286%29%29%28x-4-sqrt%286%29%29\"
\n" ); document.write( "Simplifying,
\n" ); document.write( "\"f%28x%29=%28x-2%29%28x%5E2-%282i%29%5E2%29%28%28x-4%29%5E2-%28sqrt%286%29%29%5E2%29\"
\n" ); document.write( "\"f%28x%29=%28x-2%29%28x%5E2%2B4%29%28%28x-4%29%5E2-6%29\"
\n" ); document.write( "\"f%28x%29=%28x-2%29%28x%5E2%2B4%29%28x%5E2-8x%2B16-6%29\"
\n" ); document.write( "\"f%28x%29=%28x-2%29%28x%5E2%2B4%29%28x%5E2-8x%2B10%29\"
\n" ); document.write( "If we really must, we multiply to get
\n" ); document.write( "\"f%28x%29=%28x-2%29%28x%5E4-32x%5E3%2B14x%5E2-32x%2B40%29\" , and
\n" ); document.write( "\"f%28x%29=x%5E5-10x%5E4%2B30x%5E3-60x%5E2%2B104x-80%29\" . \n" ); document.write( "
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