SOLUTION: 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. Th

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.
Thank you!
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
TIP: The conventional way to write is sqrt(6).

1) For each real or complex zero of a polynomial ,
there is a factor in the factorization of the polynomial.
So, if has , , and as zeros,
the factors , , and are in the factorization of .

2) If a polynomial has real coefficients and has a non-real complex number for example) as a zero,
the conjugate complex number, is also a zero of the polynomial,
and
if has rational coefficients and has an irrational number such as as a zero,
the conjugate irrational number, , is also a zero of the polynomial.
In other words, irrational zeros and non-real complex zeros come as conjugate pairs.
So, along with , also has as a zero,
and along with , also has as a zero.

Then, the factors and
are also in the factorization of .

3) Besides at least one factor of the form for every zero,
the factorization of a polynomial with rational coefficients,
also has a constant factor , which will be the leading coefficient.

So,

Simplifying,

If we really must, we multiply to get
, and
.
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