To find a polynomial function of the lowest degree with rational coefficients that has [tex]\( 4 + i \)[/tex] and [tex]\( 2 \)[/tex] as some of its zeros, we follow these steps:
1. Identify all roots: Since [tex]\( 4 + i \)[/tex] is a root and we're seeking a polynomial with rational coefficients, its complex conjugate [tex]\( 4 - i \)[/tex] must also be a root. Thus, the roots are [tex]\( 4 + i \)[/tex], [tex]\( 4 - i \)[/tex], and [tex]\( 2 \)[/tex].
2. Form linear factors: Using these roots, we can construct the corresponding linear factors:
- For [tex]\( 4 + i \)[/tex]: [tex]\( (x - (4 + i)) \)[/tex]
- For [tex]\( 4 - i \)[/tex]: [tex]\( (x - (4 - i)) \)[/tex]
- For [tex]\( 2 \)[/tex]: [tex]\( (x - 2) \)[/tex]
3. Multiply the factors: The polynomial is formed by multiplying these linear terms together.
First, multiply the factors corresponding to the complex roots:
[tex]\[
(x - (4 + i))(x - (4 - i))
\][/tex]
Using the difference of squares formula [tex]\( (a - b)(a + b) = a^2 - b^2 \)[/tex], where [tex]\( a = x - 4 \)[/tex] and [tex]\( b = i \)[/tex], we get:
[tex]\[
[(x - 4) - i][(x - 4) + i] = (x - 4)^2 - i^2
\][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
(x - 4)^2 - (-1) = (x - 4)^2 + 1
\][/tex]
Expanding [tex]\( (x - 4)^2 \)[/tex]:
[tex]\[
(x - 4)^2 + 1 = (x^2 - 8x + 16) + 1 = x^2 - 8x + 17
\][/tex]
4. Include the remaining root: Now, multiply by the factor corresponding to the root [tex]\( 2 \)[/tex]:
[tex]\[
(x^2 - 8x + 17)(x - 2)
\][/tex]
5. Expand the final product:
[tex]\[
x^2(x - 2) - 8x(x - 2) + 17(x - 2) = x^3 - 2x^2 - 8x^2 + 16x + 17x - 34 = x^3 - 10x^2 + 33x - 34
\][/tex]
Thus, the polynomial function in expanded form is:
[tex]\[
f(x) = \boxed{1.0x^3 - 10.0x^2 + 33.0x - 34.0}
\][/tex]
