To find a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros, we proceed as follows:
1. Identify the given zeros and their conjugates:
- The given zeros are [tex]\(3 - i\)[/tex] and [tex]\(\sqrt{6}\)[/tex].
- Since the polynomial must have rational coefficients, the complex zero [tex]\(3 - i\)[/tex] must be paired with its conjugate [tex]\(3 + i\)[/tex].
- Similarly, the irrational zero [tex]\(\sqrt{6}\)[/tex] must be paired with its conjugate [tex]\(-\sqrt{6}\)[/tex].
2. Form factors of the polynomial:
- For each pair of zeros, we form a factor of the polynomial. The factors corresponding to the given zeros and their conjugates are:
[tex]\[
(x - (3 - i))(x - (3 + i))
\][/tex]
[tex]\[
(x - \sqrt{6})(x + \sqrt{6})
\][/tex]
3. Expand the factors:
- First, we expand the factor for the complex conjugate zeros:
[tex]\[
(x - (3 - i))(x - (3 + i)) = (x - 3 + i)(x - 3 - i)
\][/tex]
[tex]\[
= ((x - 3) + i)((x - 3) - i)
\][/tex]
[tex]\[
= (x - 3)^2 - i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= (x - 3)^2 - (-1)
\][/tex]
[tex]\[
= (x - 3)^2 + 1
\][/tex]
[tex]\[
= x^2 - 6x + 9 + 1
\][/tex]
[tex]\[
= x^2 - 6x + 10
\][/tex]
- Next, we expand the factor for the irrational conjugate zeros:
[tex]\[
(x - \sqrt{6})(x + \sqrt{6})
\][/tex]
[tex]\[
= x^2 - (\sqrt{6})^2
\][/tex]
[tex]\[
= x^2 - 6
\][/tex]
4. Form the polynomial by multiplying the factors:
- Now, to get the polynomial function, we need to multiply the expanded factors together:
[tex]\[
f(x) = (x^2 - 6x + 10)(x^2 - 6)
\][/tex]
5. Expand the resulting polynomial:
[tex]\[
f(x) = (x^2 - 6x + 10)(x^2 - 6)
\][/tex]
We distribute each term of the first polynomial to each term of the second polynomial:
[tex]\[
= x^2(x^2 - 6) - 6x(x^2 - 6) + 10(x^2 - 6)
\][/tex]
[tex]\[
= (x^4 - 6x^2) - (6x^3 - 36x) + (10x^2 - 60)
\][/tex]
[tex]\[
= x^4 - 6x^2 - 6x^3 + 36x + 10x^2 - 60
\][/tex]
6. Combine like terms:
[tex]\[
= x^4 - 6x^3 + 4x^2 + 36x - 60
\][/tex]
Therefore, the polynomial function in expanded form is:
[tex]\[ f(x) = x^4 - 6x^3 + 4x^2 + 36x - 60. \][/tex]
