Final answer:
The polynomial with degree 4 and given zeros 6 (with multiplicity 2) and 2¹ is f(x) = (x - 6)² (x² + 4), where the complex and real zeros are combined. The correct answer is b) f(x) = (x - 6)² (x² + 4).
Step-by-step explanation:
To form a polynomial f(x) with real coefficients having the given degree and zeros of Degree 4; zeros: 6 (multiplicity 2), 2¹, we must consider the given zeros and their multiplicities. The fact that the zeros include a complex number 2¹ and its conjugate -2¹ implies that the polynomial will have these as factors in conjugate pairs because complex zeros of polynomials with real coefficients always come in conjugate pairs.
The correct polynomial that incorporates all these zeros is f(x) = (x - 6)² (x - 2¹)(x + 2¹), which can be simplified using the difference of squares for the complex terms (x - 2¹)(x + 2¹) = x² + 4. This gives us:
f(x) = (x - 6)² (x² + 4).
Therefore, the correct answer is b) f(x) = (x - 6)² (x² + 4).