To find the polynomial function of the lowest degree with only real coefficients and the given zeros [tex]\( \sqrt{7}, -\sqrt{7}, \)[/tex] and [tex]\( 4 \)[/tex], we can follow these steps:
1. Identify the zeros and form the factors:
The given zeros imply the factors of the polynomial. If the polynomial has zeros [tex]\( \sqrt{7}, -\sqrt{7}, \)[/tex] and [tex]\( 4 \)[/tex], the corresponding factors are [tex]\( (x - \sqrt{7}), (x + \sqrt{7}), \)[/tex] and [tex]\( (x - 4) \)[/tex].
2. Multiply the factors to form the polynomial:
Start by multiplying the factors.
First, multiply [tex]\( (x - \sqrt{7}) \)[/tex] and [tex]\( (x + \sqrt{7}) \)[/tex]:
[tex]\[
(x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7
\][/tex]
3. Multiply the result by the remaining factor:
Now multiply [tex]\( (x^2 - 7) \)[/tex] by [tex]\( (x - 4) \)[/tex]:
[tex]\[
(x^2 - 7)(x - 4) = x^3 - 4x^2 - 7x + 28
\][/tex]
4. Write the polynomial function:
The polynomial of the lowest degree with real coefficients that has the given zeros is:
[tex]\[
f(x) = x^3 - 4x^2 - 7x + 28
\][/tex]
Thus, the correct polynomial function is:
D. [tex]\( f(x) = x^3 - 4x^2 - 7x + 28 \)[/tex]
