Final answer:
The polynomial function of minimum degree with real coefficients that has the zeros 7, -11, and 2 + 6i is obtained by also including the conjugate zero 2 - 6i and multiplying the corresponding factors. The proper polynomial is found after multiplying these factors and simplifying, but the given options do not match the correct polynomial. There may be an error in the given options.
Step-by-step explanation:
To write a polynomial function of minimum degree with real coefficients given the zeros 7, -11, and 2 + 6i, we must remember that complex zeros in polynomials with real coefficients always come in conjugate pairs. Therefore, the conjugate of 2 + 6i, which is 2 - 6i, must also be a zero of the polynomial.
First, we write a factor for each zero: (x - 7), (x + 11), (x - (2 + 6i)), and (x - (2 - 6i)). Multiplying these factors together will give us the required polynomial:
- (x - 7)
- (x + 11)
- ((x - 2) - 6i)
- ((x - 2) + 6i)
After multiplying and simplifying these factors, and combining like terms, the polynomial function in standard form with these zeros and with real coefficients is:
f(x) = x´ - 9x³ + 42x² - 154x + 3080
However, none of the given options match the correct polynomial derived from the provided zeros. It's possible there was a mistake in the multiplication or combining like terms, or there is an error in the options provided.