Final answer:
The polynomial function Q of degree 4 with zeros 1 + √5 and -7i, given that it has rational coefficients, will also include their conjugates 1 - √5 and 7i. The equation for Q is found by multiplying the corresponding quadratic factors (x² - 2x - 4) and (x² + 49), resulting in Q(x) = (x² - 2x - 4)(x² + 49).
Step-by-step explanation:
A polynomial function Q of degree 4 with rational coefficients has zeros 1 + √5 and -7i. To find an equation for Q, we need to keep in mind that the polynomial will have complex conjugate pairs and real number pairs as its zeros if the coefficients are rational. The given zero 1 + √5 has its conjugate 1 - √5, and the imaginary zero -7i has its conjugate 7i.
Using the fact that the zeros of a polynomial function are the values for which the function equals zero, we can start building the factors of Q. For the real zero and its conjugate, we get a quadratic factor:
(x - (1 + √5))(x - (1 - √5)) = x² - 2x + (1² - (√5)²) = x² - 2x - 4.
For the imaginary zero and its conjugate, we get another quadratic factor:
(x - (-7i))(x - (7i)) = x² - (7i)² = x² + 49.
By multiplying these two quadratic factors together, we get the polynomial Q which will be:
Q(x) = (x² - 2x - 4)(x² + 49).
This is the equation for the polynomial function Q of degree 4 with the given zeros.